From f8018a3c5d55bde11ae43be0b05fc6455a94b52a Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Tue, 12 Sep 2023 13:52:38 +0200 Subject: [PATCH 01/41] Changed Adam Optimizer visualization to two-step process. --- docs/src/images/adam_optimizer.png | Bin 35115 -> 58818 bytes docs/src/images/adam_optimizer.tex | 2 +- docs/src/images/general_optimization.pdf | Bin 59770 -> 59754 bytes docs/src/images/general_optimization.png | Bin 44806 -> 42854 bytes docs/src/images/general_optimization.tex | 2 +- .../general_optimization_with_boundary.png | Bin 104584 -> 102813 bytes .../general_optimization_with_boundary.tex | 2 +- 7 files changed, 3 insertions(+), 3 deletions(-) diff --git a/docs/src/images/adam_optimizer.png b/docs/src/images/adam_optimizer.png index 8b5d1a8c56f8c8568e1c8adf334f6db2f01203b7..62569f3110be749b11ac5a62349d319c7b3ce50b 100644 GIT binary patch literal 58818 zcmbTe1yomC`z?&eLM&85K}AHRL8YWa1O*A{Mo~aoy1@X&KrBE&QaVJWqzzE%25CjQ 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zs=D%#T#4g31_Ty(EMYA{9*|pDNEjUSIZz z9oRKNNj#?cL`)Ihy}KLzII_nlf6X1Nl%%$nkva79kiQ8ouQ*(~G&MG6@8A%wiqotr zkjL4ecq1I9NCVIs4Hq5!{qN3s0a*%)Rm5nBnx?;e2_TPW0ThixF$O3MG3wvRX#-^? zr#SINQfANG>WwpJdH~fQ022GpmNo^ZN*trbK#s8L(Isg9JD)Z#3=KD0PAqN?K0AU- z{ojK_EFKfK{^uYFn*Tli+t~ko;lI!S?|<9=UGo3;hXekfU*z~Ny!`iF@_(=8zyJCF uzaLBYU)cQbyJWbH|9dF@|Mo|f{;<=Y_vJdWwiZTQg_hbW)iUL?f&T}J@YiGj diff --git a/docs/src/images/general_optimization_with_boundary.tex b/docs/src/images/general_optimization_with_boundary.tex index 9ce0295e9..a576864d9 100644 --- a/docs/src/images/general_optimization_with_boundary.tex +++ b/docs/src/images/general_optimization_with_boundary.tex @@ -31,7 +31,7 @@ \draw[->] (TYM) -- (ghorY) node[pos=.5, left] {$\Omega$}; \draw[->, mred] (ghorY) -- (ghor); - \draw[->] (ghor) -- (ghor2) node[pos=.5, sloped, below] {Adam}; + \draw[->] (ghor) -- (cache) node[pos=.5, sloped, below] {Adam}; \draw[->] (cache) -- (ghor2) node[pos=.5, sloped, above] {Adam}; \draw[->] (ghor2) -- (M) node[pos=.5, right] {Retraction}; \draw[->, mgreen] (M2) -- (Euc) node[pos=.5, left] {\color{mgreen}AD}; From 107e27f2e53d730f07b728b61f7eb41c38040d60 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Tue, 12 Sep 2023 17:18:46 +0200 Subject: [PATCH 02/41] Added description of multihead attention layer. --- docs/src/layers/multihead_attention_layer.md | 22 +++++++++++++++++++- 1 file changed, 21 insertions(+), 1 deletion(-) diff --git a/docs/src/layers/multihead_attention_layer.md b/docs/src/layers/multihead_attention_layer.md index ac7444126..5c2f8f524 100644 --- a/docs/src/layers/multihead_attention_layer.md +++ b/docs/src/layers/multihead_attention_layer.md @@ -1,7 +1,27 @@ # Multihead Attention Layer -In order to arrive from the [attention layer](attention_layer.md) at the **multihead attention layer** we only have to do a simple modification: +In order to arrive from the [attention layer](attention_layer.md) at the **multihead attention layer** we have to do a few modifications: +Note that these neural networks were originally developed for natural language processing (NLP) tasks and the terminology used here bears some resemblance to that field. +The input to a multihead attention layer typicaly comprises three components: + +1. Values $V\in\mathbb{R}^{n\times{}T}$: a matrix whose columns are **value vectors**, +2. Queries $Q\in\mathbb{R}^{n\times{}T}$: a matrix whose columns are **query vectors**, +3. Keys $K\in\mathbb{R}^{n\times{}T}$: a matrix whose columns are **key vectors**. + +Regular attention performs the following operation: + +```math +\mathrm{Attention}(Q,K,V) = V\mathrm{softmax}(\frac{K^TQ}{\sqrt{n}}), +``` + +where $n$ is the dimension of the vectors in $V$, $Q$ and $K$. The softmax activation function here acts column-wise, so it can be seen as a transformation $\mathrm{softmax}:\mathbb{R}^{T}\to\mathbb{R}^T$ with $[\mathrm{softmax}(v)]_i = e^{v_i}/\left(\sum_{j=1}e^{v_j}\right)$. The $K^TQ$ term is a similarity matrix between the queries and the vectors. + +The transformer contains a **self-attention mechanism**, i.e. takes an input $X$ and then transforms it linearly to $V$, $Q$ and $K$, i.e. $V = P^VX$, $Q = P^QX$ and $K = P^KX$. What distinguishes the multihead attention layer from the singlehead attention layer, is that there is not just one $P^V$, $P^Q$ and $P^K$, but there are several: one for each **head** of the multihead attention layer. After computing the individual values, queries and vectors, and after applying the softmax, the outputs are then concatenated together in order to obtain again an array that is of the same size as the input array: + +![](../images/mha.png) + +Here the various $P$ matrices can be interpreted as being projections onto lower-dimensional subspaces, hence the designation by the letter $P$. Because of this interpretation as projection matrices onto smaller spaces that should **capture features in the input data** it makes sense to constrain these elements to be part of the Stiefel manifold. ## References - Vaswani, Ashish, et al. "Attention is all you need." Advances in neural information processing systems 30 (2017). \ No newline at end of file From d658341ae8860a04068be97f23c45e5e01b8140d Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Tue, 12 Sep 2023 17:19:56 +0200 Subject: [PATCH 03/41] Got rid of the assign_matrix and assign_tensor functions. Hope this still works. --- src/kernels/inverses/inverse_kernel.jl | 14 ++++++++++---- 1 file changed, 10 insertions(+), 4 deletions(-) diff --git a/src/kernels/inverses/inverse_kernel.jl b/src/kernels/inverses/inverse_kernel.jl index de65b0770..e39af406e 100644 --- a/src/kernels/inverses/inverse_kernel.jl +++ b/src/kernels/inverses/inverse_kernel.jl @@ -60,11 +60,17 @@ end function tensor_inverse(A::AbstractArray{T, 3}) where T A_inv = zero(A) total_length = size(A, 3) - for k in 1:total_length - B = assign_matrix(A, k) - B_inv = B\one(B) - A_inv += assign_tensor(B_inv, total_length, k) + #for k in 1:total_length + # B = assign_matrix(A, k) + # B_inv = B\one(B) + # A_inv += assign_tensor(B_inv, total_length, k) + #end + + for k in axes(A_inv, 3) + B = @view A[:,:,k] + A_inv[:,:,k] .= B \ one(B) end + A_inv end From 7061da16318b0b64c0a3692b6020355fc073c982 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Tue, 12 Sep 2023 17:20:52 +0200 Subject: [PATCH 04/41] Option to use LogExpFunctions.softmax => doesn't seem to be working. --- src/GeometricMachineLearning.jl | 9 +++------ 1 file changed, 3 insertions(+), 6 deletions(-) diff --git a/src/GeometricMachineLearning.jl b/src/GeometricMachineLearning.jl index 05db28693..a486ab8e2 100644 --- a/src/GeometricMachineLearning.jl +++ b/src/GeometricMachineLearning.jl @@ -35,6 +35,7 @@ module GeometricMachineLearning export dim import GeometricIntegrators.Integrators: method import NNlib: σ, sigmoid, softmax + #import LogExpFunctions: softmax export CPU, GPU export Chain, NeuralNetwork @@ -354,9 +355,5 @@ module GeometricMachineLearning export integrate, integrate_step! include("integrator/sympnet_integrator.jl") - - - - - -end + +end \ No newline at end of file From d460f00484a73d19d86b9c56daf67515dddb1891 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Tue, 12 Sep 2023 17:23:08 +0200 Subject: [PATCH 05/41] Included this rescaling factor in the computation of the softmax. --- src/layers/multi_head_attention.jl | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/src/layers/multi_head_attention.jl b/src/layers/multi_head_attention.jl index ad4612f14..0ae8757f6 100644 --- a/src/layers/multi_head_attention.jl +++ b/src/layers/multi_head_attention.jl @@ -82,7 +82,7 @@ function (d::MultiHeadAttention{M, M, Stiefel, Retraction, true})(x::AbstractMat output = typeof(x)(zeros(T, 0, input_length)) for i in 1:d.n_heads key = Symbol("head_"*string(i)) - output = vcat(output, ps.PV[key]'*x*Lux.softmax((ps.PQ[key]'*x)'*(ps.PK[key]'*x))) + output = vcat(output, ps.PV[key]'*x*softmax((ps.PQ[key]'*x)'*(ps.PK[key]'*x)/T(sqrt(dim)))) end x + output end @@ -94,7 +94,7 @@ function (d::MultiHeadAttention{M, M, Stiefel, Retraction, false})(x::AbstractMa output = similar(x, 0, input_length) for i in 1:d.n_heads key = Symbol("head_"*string(i)) - output = vcat(output, ps.PV[key]'*x*Lux.softmax((ps.PQ[key]'*x)'*(ps.PK[key]'*x))) + output = vcat(output, ps.PV[key]'*x*softmax((ps.PQ[key]'*x)'*(ps.PK[key]'*x)/T(sqrt(dim)))) end output end @@ -123,7 +123,7 @@ function (d::MultiHeadAttention{M, M, Stiefel, Retraction, true})(x::AbstractArr V_tensor = mat_tensor_mul(ps.PV[key]', x) QK_tensor = tensor_transpose_tensor_mul(Q_tensor, K_tensor) - single_head_output = tensor_tensor_mul(V_tensor, Lux.softmax(QK_tensor)) + single_head_output = tensor_tensor_mul(V_tensor, softmax(QK_tensor/T(sqrt(dim)))) output = vcat(output, single_head_output) # KernelAbstractions.synchronize(backend) end @@ -151,7 +151,7 @@ function (d::MultiHeadAttention{M, M, Stiefel, Retraction, false})(x::AbstractAr V_tensor = mat_tensor_mul(ps.PV[key]', x) QK_tensor = tensor_transpose_tensor_mul(Q_tensor, K_tensor) - single_head_output = tensor_tensor_mul(V_tensor, Lux.softmax(QK_tensor)) + single_head_output = tensor_tensor_mul(V_tensor, softmax(QK_tensor/T(sqrt(dim)))) output = vcat(output, single_head_output) # KernelAbstractions.synchronize(backend) end From 86fc7243af95264b61c11bd1cc1b33ff950fd722 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Tue, 12 Sep 2023 17:50:25 +0200 Subject: [PATCH 06/41] Now performing two plots (as in the paper). --- scripts/transformer_new.jl | 27 +++++++++++++++++++-------- 1 file changed, 19 insertions(+), 8 deletions(-) diff --git a/scripts/transformer_new.jl b/scripts/transformer_new.jl index 6570a77f5..d0cf0406f 100644 --- a/scripts/transformer_new.jl +++ b/scripts/transformer_new.jl @@ -15,11 +15,11 @@ image_dim = 28 patch_length = 7 transformer_dim = 49 n_heads = 7 -n_layers = 16 +n_layers = 10 number_of_patch = (image_dim÷patch_length)^2 -batch_size = 512 +batch_size = 2048 activation = softmax -n_epochs = 20 +n_epochs = 1000 add_connection = false backend = CUDABackend() @@ -54,25 +54,36 @@ function transformer_training(Ψᵉ::Chain; backend=CPU(), n_training_steps=1000 progress_object = Progress(n_training_steps; enabled=true) + loss_array = zeros(eltype(train_x), n_training_steps) for i in 1:n_training_steps redraw_batch!(dl) + # ask Michael to take a look at this. Probably not good for performance. loss_val, pb = Zygote.pullback(ps -> loss(Ψᵉ, ps, dl), ps) dp = pb(one(loss_val))[1] optimization_step!(optimizer_instance, Ψᵉ, ps, dp) ProgressMeter.next!(progress_object; showvalues = [(:TrainingLoss, loss_val)]) + loss_array[i] = loss_val end println("final test loss: ", loss(Ψᵉ, ps, dl_test), "\n") - ps + loss_array, ps end # calculate number of epochs n_training_steps = Int(ceil(length(train_y)*n_epochs/batch_size)) -ps₁ = transformer_training(model1, backend=backend, n_training_steps=n_training_steps) -ps₂ = transformer_training(model2, backend=backend, n_training_steps=n_training_steps) +loss_array2, ps2 = transformer_training(model2, backend=backend, n_training_steps=n_training_steps) +loss_array1, ps1 = transformer_training(model1, backend=backend, n_training_steps=n_training_steps) +loss_array3, ps3 = transformer_training(model2, backend=backend, n_training_steps=n_training_steps, o=GradientOptimizer(0.001)) +loss_array4, ps4 = transformer_training(model2, backend=backend, n_training_steps=n_training_steps, o=MomentumOptimizer(0.001, 0.5)) -#loss_array₃ = transformer_training(Ψᵉ₂, batch_size, training_steps, err_freq, StandardOptimizer(0.001)) -#loss_array₄ = transformer_training(Ψᵉ₂, batch_size, training_steps, err_freq, MomentumOptimizer(0.001, 0.5)) +p1 = plot(loss_array1, color=1, label="Regular weights", ylimits=(0.,1.4)) +plot!(p1, loss_array2, color=2, label="Weights on Stiefel Manifold") +png(p1, "Stiefel_Regular") + +p2 = plot(loss_array2, color=2, label="Adam", ylimits=(0.,1.4)) +plot!(p2, loss_array3, color=1, label="Gradient") +plot!(p2, loss_array4, color=3, label="Momentum") +png(p2, "Adam_Gradient_Momentum") \ No newline at end of file From 4613a02b434e0aea252e6ace62309ceda0a3d20c Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 10:54:53 +0200 Subject: [PATCH 07/41] Added a tutorial and started describing the data loader. --- docs/src/data_loader/data_loader.md | 8 ++++ docs/tutorials/mnist_tutorial.md | 59 +++++++++++++++++++++++++++++ 2 files changed, 67 insertions(+) create mode 100644 docs/src/data_loader/data_loader.md create mode 100644 docs/tutorials/mnist_tutorial.md diff --git a/docs/src/data_loader/data_loader.md b/docs/src/data_loader/data_loader.md new file mode 100644 index 000000000..1e60c1804 --- /dev/null +++ b/docs/src/data_loader/data_loader.md @@ -0,0 +1,8 @@ +# Data Loader + +`GeometricMachineLearning` provides flexible routines to load and manage data for training neural networks. +`DataLoader` has several constructors: + +1. If provided with a tensor, then it assumes the first axis is the system dimension, the second axis is the dimension of the parameter space, and the third axis gives the time evolution of the system. + +2. If provided with a tensor and a vector, it assumes the data are related to a classification task. \ No newline at end of file diff --git a/docs/tutorials/mnist_tutorial.md b/docs/tutorials/mnist_tutorial.md new file mode 100644 index 000000000..0527f7cce --- /dev/null +++ b/docs/tutorials/mnist_tutorial.md @@ -0,0 +1,59 @@ +# MNIST tutorial + +This is a short tutorial that shows how we can use `GeometricMachineLearning` to build a vision transformer and apply it for MNIST, while also putting some of the weights on a manifold. + +First, we need to import the relevant packages: + +```julia +using GeometricMachineLearning, CUDA +import Zygote, MLDatasets +``` + +In this example `Zygote` as an AD routine and we get the dataset from `MLDatasets`. First we need to load the data set, and put it on GPU (if you have one): + +```julia +train_x, train_y = MLDatasets.MNIST(split=:train)[:] +test_x, test_y = MLDatasets.MNIST(split=:test)[:] +train_x = train_x |> cu +test_x = test_x |> cu +train_y = train_y |> cu +test_y = test_y |> cu +``` + +`GeometricMachineLearning` has built-in data loaders that make it particularly easy to handle data: + +```julia +patch_length = 7 +dl = DataLoader(train_x, train_y, batch_size=512, patch_length=patch_length) +dl_test = DataLoader(train_x, train_y, batch_size=length(y), patch_length=patch_length) +``` + +The second line in the above code snippet indicates that we use the entire data set as one "batch" when processing the test set. For training, the batch size was here set to 512. + +```julia +ps = initialparameters(backend, eltype(dl.data), Ψᵉ) + +optimizer_instance = Optimizer(o, ps) + +println("initial test accuracy: ", accuracy(Ψᵉ, ps, dl_test), "\n") + +progress_object = Progress(n_training_steps; enabled=true) + +loss_array = zeros(eltype(train_x), n_training_steps) +for i in 1:n_training_steps + redraw_batch!(dl) + # get rid of try catch statement. This softmax issue should be solvable! + loss_val, pb = try Zygote.pullback(ps -> loss(Ψᵉ, ps, dl), ps) + catch + loss_array[i] = loss_array[i-1] + continue + end + dp = pb(one(loss_val))[1] + + optimization_step!(optimizer_instance, Ψᵉ, ps, dp) + ProgressMeter.next!(progress_object; showvalues = [(:TrainingLoss, loss_val)]) + loss_array[i] = loss_val +end + +println("final test accuracy: ", accuracy(Ψᵉ, ps, dl_test), "\n") +``` \ No newline at end of file From 955b005f9611bd51bf55db21d6bc3e6d6e186d49 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 11:01:38 +0200 Subject: [PATCH 08/41] Added the new files. --- docs/make.jl | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/docs/make.jl b/docs/make.jl index 5d262ff3d..651594017 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -36,6 +36,12 @@ makedocs(; "Attention" => "layers/attention_layer.md", "Multihead Attention" => "layers/multihead_attention_layer.md", ], + "Data Loader" =>[ + "Routines" => "data_loader/data_loader.md", + ], + "Tutorials" =>[ + "MNIST" => "../tutorials/mnist_tutorial.md", + ], "Library" => "library.md", ], ) From 1e936726f0719ae95703530f7e70c75a2a4a5c43 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 14:12:28 +0200 Subject: [PATCH 09/41] Copied the Lux pipeline for now. --- .buildkite/pipeline.yml | 68 +++++++++++++++++++++++++++++++++++++++++ 1 file changed, 68 insertions(+) create mode 100644 .buildkite/pipeline.yml diff --git a/.buildkite/pipeline.yml b/.buildkite/pipeline.yml new file mode 100644 index 000000000..1096178d9 --- /dev/null +++ b/.buildkite/pipeline.yml @@ -0,0 +1,68 @@ +steps: + - label: ":julia: Julia {{matrix.julia}} + CUDA GPU" + plugins: + - JuliaCI/julia#v1: + version: "{{matrix.julia}}" + - JuliaCI/julia-test#v1: + test_args: "--quickfail" + - JuliaCI/julia-coverage#v1: + codecov: true + dirs: + - src + - ext + agents: + queue: "juliagpu" + cuda: "*" + env: + GROUP: "CUDA" + if: build.message !~ /\[skip tests\]/ + timeout_in_minutes: 240 + matrix: + setup: + julia: + - "1" + - "1.6" + - "nightly" + adjustments: + - with: + julia: "1.6" + soft_fail: true + - with: + julia: "nightly" + soft_fail: true + - label: "Documentation" + plugins: + - JuliaCI/julia#v1: + version: "1" + - JuliaCI/julia-coverage#v1: + codecov: true + dirs: + - src + - ext + command: | + julia --project --code-coverage=user --color=yes --threads=3 -e ' + println("--- :julia: Instantiating project") + using Pkg + Pkg.instantiate() + Pkg.activate("docs") + Pkg.develop(PackageSpec(path=pwd())) + Pkg.instantiate() + println("+++ :julia: Building documentation") + + Pkg.activate("docs") + include("docs/tutorials.jl") + Pkg.activate("docs") + include("docs/make.jl")' + agents: + queue: "juliagpu" + cuda: "*" + env: + DATADEPS_ALWAYS_ACCEPT: true + JULIA_DEBUG: "Documenter" + GKSwstype: "100" # https://discourse.julialang.org/t/generation-of-documentation-fails-qt-qpa-xcb-could-not-connect-to-display/60988 + if: build.message !~ /\[skip docs\]/ && !build.pull_request.draft + timeout_in_minutes: 240 + +env: + SECRET_CODECOV_TOKEN: "jQ0BMTQgyZx7QGyU0Q2Ec7qB9mtE2q/tDu0FsfxvEG7/zOAGvXkyXrzIFFOQxvDoFcP+K2+hYZKMxicYdNqzr5wcxu505aNGN2GM3wyegAr+hO6q12bCFYx6qXzU9FLCCdeqINqn9gUSSOlGtWNFrbAlrTyz/D4Yo66TqBDzvaLL63FMnhCLaXW/zJt3hNuEAJaPY2O6Ze1rX2WZ3Y+i+s3uQ8aLImtoCJhPe8CRx+OhuYiTzGhynFfGntZ0738/1RN4gNM0S/hTC4gLE7XMVBanJpGh32rFaiDwW4zAyXKBrDkL3QA3MS1RvLTJxGJ085S16hCk0C4ddAhZCvIM9Q==;U2FsdGVkX1+bXdFeKMs5G79catOCyby2n07A2fg0FjVAvrjQLZ0yfvDS4paJiFikLkodho0khz2YALKb2Y0K6w==" + SECRET_DOCUMENTER_KEY: "iRC4P/r5o9pARB670eK9jPlKQKgkTMDAyvp2GbLG8WwLuT8T1VcWx/o4+ofGlzbTh5Z+LuFgPXfgqkjGuoWLcocHNm78xQMNMywB4rcLB2shqp8xG2vhglgnTBBS4EiyPAtVqGyi5AKmfF95PfkJvnI0Lqg5P/RWQvNGywLAR0Ikgr/lqocm2CvkFGbpMzpGxGvj76JYOusVeKvGAp698TXqPabSZR2oZQLfYnEZnaO8ivkqvMGQSXfgzoIMjCOrN1rSa84SWeI9BDeBslzDHwaYGlvjpfCyviiLtKj4t5Acl1gVE0qxxZxWuALIU6z+C1W8TbW7ZDCBUFs6UTIT+Q==;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" \ No newline at end of file From dd10a62b40dbb097ac6119db5258cca9b8c14ba7 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 14:13:43 +0200 Subject: [PATCH 10/41] Started adding comments on the Cayley transform and the geodesic retraction. --- docs/src/optimizers/manifold_related/cayley.md | 3 +++ docs/src/optimizers/manifold_related/geodesic.md | 3 +++ 2 files changed, 6 insertions(+) create mode 100644 docs/src/optimizers/manifold_related/cayley.md create mode 100644 docs/src/optimizers/manifold_related/geodesic.md diff --git a/docs/src/optimizers/manifold_related/cayley.md b/docs/src/optimizers/manifold_related/cayley.md new file mode 100644 index 000000000..de829ec1e --- /dev/null +++ b/docs/src/optimizers/manifold_related/cayley.md @@ -0,0 +1,3 @@ +# The Cayley Retraction + +The Cayley transformation is one of the most popular retractions. For several matrix Lie groups it is a mapping from the Lie algebra $\mathfrak{g}$ onto the Lie group $G$. \ No newline at end of file diff --git a/docs/src/optimizers/manifold_related/geodesic.md b/docs/src/optimizers/manifold_related/geodesic.md new file mode 100644 index 000000000..9ff9d645c --- /dev/null +++ b/docs/src/optimizers/manifold_related/geodesic.md @@ -0,0 +1,3 @@ +# Geodesic Retraction + +General **retractions** are approximations of the exponential map. In `GeometricMachineLearning` we can, instead of using an approximation, solve the geodesic equation exactly (up to numerical error) by specifying `Geodesic()` as the argument of layers that have manifold weights. \ No newline at end of file From 296247e8ca2f8cd2fec4452f03d7f0629a39163e Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 15:42:53 +0200 Subject: [PATCH 11/41] Added computation of cayley function we are using. --- .../src/optimizers/manifold_related/cayley.md | 43 ++++++++++++++++++- 1 file changed, 42 insertions(+), 1 deletion(-) diff --git a/docs/src/optimizers/manifold_related/cayley.md b/docs/src/optimizers/manifold_related/cayley.md index de829ec1e..febbc4ce2 100644 --- a/docs/src/optimizers/manifold_related/cayley.md +++ b/docs/src/optimizers/manifold_related/cayley.md @@ -1,3 +1,44 @@ # The Cayley Retraction -The Cayley transformation is one of the most popular retractions. For several matrix Lie groups it is a mapping from the Lie algebra $\mathfrak{g}$ onto the Lie group $G$. \ No newline at end of file +The Cayley transformation is one of the most popular retractions. For several matrix Lie groups it is a mapping from the Lie algebra $\mathfrak{g}$ onto the Lie group $G$. +They Cayley retraction reads: + +```math + \mathrm{Cayley}(C) = \left(\mathbb{I} -\frac{1}{2}C\right)^{-1}\left(\mathbb{I} +\frac{1}{2}C\right). +``` +This is easily checked to be a retraction, i.e. $\mathrm{Cayley}(\mathbb{O}) = \mathbb{I}$ and $\frac{\partial}{\partial{}t}\mathrm{Cayley}(tC) = C$. + +What we need in practice is not the computation of the Cayley transform of an arbitrary matrix, but the Cayley transform of an element of $\mathfrak{g}^\mathrm{hor}$, the [global tangent space representation](../../arrays/stiefel_lie_alg_horizontal.md). + +The elements of $\mathfrak{g}^\mathrm{hor}$ can be written as: + +```math +C = \begin{bmatrix} + A & -B^T \\ + B & \mathbb{O} +\end{bmatrix} = \begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} \begin{bmatrix} \mathbb{I} & \mathbb{O} \\ \frac{1}{2}A & -B^T \end{bmatrix}, +``` + +where the second expression exploits the sparse structure of the array, i.e. it is a multiplication of a $N\times2n$ with a $2n\times{}N$ matrix. We can hence use the **Sherman-Morrison-Woodbury formula**: + +```math +(\mathbb{I} - UV)^{-1} = \mathbb{I} + U(\mathbb{I} - VU)^{-1}V +``` + +So What we have to invert is the term + +```math +\mathbb{I} - \begin{bmatrix} \mathbb{I} & \mathbb{O} \\ \frac{1}{2}A & -B^T \end{bmatrix}\begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} = +\begin{bmatrix} \mathbb{I} - \frac{1}{2}A & - \mathbb{I} \\ B^TB - \frac{1}{4}A^2 & \mathbb{I} - \frac{1}{2}A \end{bmatrix}. +``` + +The whole cayley transform is then: + +$$ +\begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} \begin{bmatrix} \mathbb{I} - \frac{1}{2}A & - \mathbb{I} \\ B^TB - \frac{1}{4}A^2 & \mathbb{I} - \frac{1}{2}A \end{bmatrix}^{-1} \begin{bmatrix} \mathbb{I} & \mathbb{O} \\ \frac{1}{2}A & -B^T \end{bmatrix}\left( E + \begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} \begin{bmatrix} \mathbb{I} \\ \frac{1}{2}A \end{bmatrix}\ \right) = \\ + +\begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} \begin{bmatrix} \mathbb{I} - \frac{1}{2}A & - \mathbb{I} \\ B^TB - \frac{1}{4}A^2 & \mathbb{I} - \frac{1}{2}A \end{bmatrix}^{-1}\left( \begin{bmatrix} \mathbb{I} \\ \frac{1}{2}A \end{bmatrix} + \begin{bmatrix} A \\ \frac{1}{2}A^2 - B^TB \end{bmatrix} \right). +$$ + + +Note that for computational reason we compute $\mathrm{Cayley}(C)E$ instead of just the Cayley transform (see the section on [retractions](retractions.md)). \ No newline at end of file From 9a48c5b3f02c24cba296a45c6b9c2c399585dab6 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 15:43:29 +0200 Subject: [PATCH 12/41] Made Cayley retraction work. --- src/optimizers/manifold_related/retractions.jl | 11 +++++++---- 1 file changed, 7 insertions(+), 4 deletions(-) diff --git a/src/optimizers/manifold_related/retractions.jl b/src/optimizers/manifold_related/retractions.jl index 9966f12ea..60cac6e0d 100644 --- a/src/optimizers/manifold_related/retractions.jl +++ b/src/optimizers/manifold_related/retractions.jl @@ -71,12 +71,15 @@ function cayley(B::StiefelLieAlgHorMatrix{T}) where T N, n = B.N, B.n E = typeof(B.B)(StiefelProjection(N, n, T)) unit = typeof(B.B)(I(n)) - unit2 = I(2*n) - exponent = unit2 - T(.5)*hcat(vcat(T(.5)*B.A*one(B.A), T(.25)*B.A*B.A - B.B'*B.B), vcat(unit, T(.5)*B.A*one(B.A))) + A_mat = B.A*one(B.A) + A_mat2 = B.A*B.A + BB = B.B'*B.B + + exponent = hcat(vcat(unit - T(.5)*A_mat, BB - T(.25)*A_mat2), vcat(-unit, unit - T(.5)*A_mat)) StiefelManifold( - (One(N, T) + T(.5)*B)* + hcat(vcat(T(.5)*A_mat, B.B), vcat(unit, zero(B.B)))* ( - E + hcat(vcat(T(.25)*B.A*one(B.A), T(.5)*B.B), vcat(T(0.5)*unit, zero(B.B)))*(exponent \ vcat(unit, T(0.5)*B.A*one(B.A))) + exponent \ (vcat(unit, T(0.5)*A_mat) + vcat(A_mat, T(0.5)*A_mat2 - BB)) ) ) end \ No newline at end of file From 07626570815e78be0882f035862e66cf654dd582 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 15:45:16 +0200 Subject: [PATCH 13/41] Added an option to output the test accuracy (for image classification tasks). --- src/GeometricMachineLearning.jl | 2 +- src/data_loader/data_loader.jl | 12 +++++++++++- 2 files changed, 12 insertions(+), 2 deletions(-) diff --git a/src/GeometricMachineLearning.jl b/src/GeometricMachineLearning.jl index a486ab8e2..503bcbbee 100644 --- a/src/GeometricMachineLearning.jl +++ b/src/GeometricMachineLearning.jl @@ -70,7 +70,7 @@ module GeometricMachineLearning include("data_loader/tensor_assign.jl") include("data_loader/data_loader.jl") include("data_loader/mnist_utils.jl") - export DataLoader, redraw_batch!, loss, onehotbatch + export DataLoader, redraw_batch!, loss, onehotbatch, accuracy # this defines empty retraction type structs (doesn't rely on anything) include("optimizers/manifold_related/retraction_types.jl") diff --git a/src/data_loader/data_loader.jl b/src/data_loader/data_loader.jl index fbdb21d8f..e67005418 100644 --- a/src/data_loader/data_loader.jl +++ b/src/data_loader/data_loader.jl @@ -72,4 +72,14 @@ function loss(model::Union{Chain, AbstractExplicitLayer}, ps::Union{Tuple, Named batch_output = model(dl.batch, ps) output_estimate = assign_output_estimate(batch_output, dl.output_size) norm(dl.output - output_estimate)/T(sqrt(dl.batch_size))/T(sqrt(dl.output_size)) -end \ No newline at end of file +end + +function accuracy(model::Chain, ps::Tuple, dl::DataLoader{T, AT, BT}) where {T, T2<:Integer, AT<:AbstractArray{T}, BT<:AbstractArray{T2}} + output_tensor = model(dl.batch, ps) + output_estimate = assign_output_estimate(output_tensor, dl.output_size) + backend = KernelAbstractions.get_backend(output_estimate) + tensor_of_maximum_elements = KernelAbstractions.zeros(backend, T2, size(output_estimate)...) + ind = argmax(output_estimate, dims=1) + tensor_of_maximum_elements[ind] .= T2(1) + (size(dl.output, 3)-sum(abs.(dl.output - tensor_of_maximum_elements))/T2(2))/size(dl.output, 3) +end From 074aaed1ac1ae7eb5957e857f0bcecae7952b8c5 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 15:45:57 +0200 Subject: [PATCH 14/41] Added new files for cayley and geodesic. --- docs/make.jl | 2 ++ 1 file changed, 2 insertions(+) diff --git a/docs/make.jl b/docs/make.jl index 651594017..053216766 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -30,6 +30,8 @@ makedocs(; "Horizontal Lift" => "optimizers/manifold_related/horizontal_lift.md", "Global Sections" => "optimizers/manifold_related/global_sections.md", "Retractions" => "optimizers/manifold_related/retractions.md", + "Geodesic Retraction" => "optimizers/manifold_related/geodesic.md", + "Cayley Retraction" => "optimizers/manifold_related/cayley.md", "Adam Optimizer" => "optimizers/adam_optimizer.md", ], "Special Neural Network Layers" => [ From f5ccd9fb10d25474d7744b48db943da9621e331d Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 15:46:43 +0200 Subject: [PATCH 15/41] Added description of general retractions. --- docs/src/optimizers/manifold_related/retractions.md | 11 +++++++++++ 1 file changed, 11 insertions(+) diff --git a/docs/src/optimizers/manifold_related/retractions.md b/docs/src/optimizers/manifold_related/retractions.md index d3ac6a886..59651c86e 100644 --- a/docs/src/optimizers/manifold_related/retractions.md +++ b/docs/src/optimizers/manifold_related/retractions.md @@ -1,5 +1,16 @@ # Retractions +## Classical Definition +Classically, retractions are defined as maps smooth maps + +```math +R: T\mathcal{M}\to\mathcal{M}:(x,v)\mapsto{}R_x(v) +``` + +such that each curve $c(t) := R_x(tv)$ satisfies $c(0) = x$ and $c'(0) = v$. + +## In `GeometricMachineLearning` + Retractions are a map from the **horizontal component** of the Lie algebra $\mathfrak{g}^\mathrm{hor}$ to the respective manifold. For optimization in neural networks (almost always first order) we solve a gradient flow equation From 0e4d9954f60ea137fce6143e83ce7b4f12b58b61 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 15:47:06 +0200 Subject: [PATCH 16/41] Added test for cayley retraction. --- test/optimizers/optimizer_convergence_tests/psd_optim.jl | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) diff --git a/test/optimizers/optimizer_convergence_tests/psd_optim.jl b/test/optimizers/optimizer_convergence_tests/psd_optim.jl index 499ce1119..ea9b6f2b5 100644 --- a/test/optimizers/optimizer_convergence_tests/psd_optim.jl +++ b/test/optimizers/optimizer_convergence_tests/psd_optim.jl @@ -27,7 +27,7 @@ A = [ 0.06476993260924702 0.8369280855305259 0.6245358125914054 0.140729967064 0.7317681833003449 0.9051355184962627 0.3376918522349117 0.436545092402125 0.3462196925686055 ] -function svd_test(A, n, train_steps=1000, tol=1e-1) +function svd_test(A, n, train_steps=1000, tol=1e-1, retraction=Cayley()) N2 = size(A,1) @assert iseven(N2) N = N2÷2 @@ -36,7 +36,7 @@ function svd_test(A, n, train_steps=1000, tol=1e-1) U_result = hcat(vcat(U_result_sing, zero(U_result_sing)), vcat(zero(U_result_sing), U_result_sing)) err_best = norm(A - U_result*U_result'*A) - model = Chain(PSDLayer(2*N, 2*n), PSDLayer(2*n, 2*N)) + model = Chain(PSDLayer(2*N, 2*n, retraction=retraction), PSDLayer(2*n, 2*N, retraction=retraction)) ps = initialparameters(CPU(), Float64, model) o₁ = Optimizer(GradientOptimizer(0.01), ps) @@ -68,4 +68,6 @@ function train_network!(o::Optimizer, model::Chain, ps::Tuple, A::AbstractMatrix ps[1].weight, ps[2].weight, error(ps) end -svd_test(A, 4) \ No newline at end of file +for retraction in (Geodesic(), Cayley()) + svd_test(A, 4, retraction=retraction) +end \ No newline at end of file From c388f1c47f920bea1211b8620341b050bbfb7d7d Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 15:51:36 +0200 Subject: [PATCH 17/41] Added Cayley retraction to test. --- test/optimizers/optimizer_convergence_tests/psd_optim.jl | 2 +- test/optimizers/optimizer_convergence_tests/svd_optim.jl | 8 +++++--- 2 files changed, 6 insertions(+), 4 deletions(-) diff --git a/test/optimizers/optimizer_convergence_tests/psd_optim.jl b/test/optimizers/optimizer_convergence_tests/psd_optim.jl index ea9b6f2b5..e85c4a94c 100644 --- a/test/optimizers/optimizer_convergence_tests/psd_optim.jl +++ b/test/optimizers/optimizer_convergence_tests/psd_optim.jl @@ -27,7 +27,7 @@ A = [ 0.06476993260924702 0.8369280855305259 0.6245358125914054 0.140729967064 0.7317681833003449 0.9051355184962627 0.3376918522349117 0.436545092402125 0.3462196925686055 ] -function svd_test(A, n, train_steps=1000, tol=1e-1, retraction=Cayley()) +function svd_test(A, n, train_steps=1000, tol=1e-1; retraction=Cayley()) N2 = size(A,1) @assert iseven(N2) N = N2÷2 diff --git a/test/optimizers/optimizer_convergence_tests/svd_optim.jl b/test/optimizers/optimizer_convergence_tests/svd_optim.jl index 92e2c23c3..165817461 100644 --- a/test/optimizers/optimizer_convergence_tests/svd_optim.jl +++ b/test/optimizers/optimizer_convergence_tests/svd_optim.jl @@ -23,13 +23,13 @@ A = [ 0.06476993260924702 0.8369280855305259 0.6245358125914054 0.140729967064 0.7317681833003449 0.9051355184962627 0.3376918522349117 0.436545092402125 0.3462196925686055 ] -function svd_test(A, n, train_steps=1000, tol=1e-1) +function svd_test(A, n, train_steps=1000, tol=1e-1; retraction=Cayley()) N = size(A,1) U, Σ, Vt = svd(A) U_result = U[:, 1:n] err_best = norm(A - U_result*U_result'*A) - model = Chain(StiefelLayer(N, n), StiefelLayer(n, N)) + model = Chain(StiefelLayer(N, n, retraction=retraction), StiefelLayer(n, N, retraction=retraction)) ps = initialparameters(CPU(), Float64, model) o₁ = Optimizer(GradientOptimizer(0.01), ps) @@ -60,4 +60,6 @@ function train_network!(o::Optimizer, model::Chain, ps::Tuple, A::AbstractMatrix ps[1].weight, ps[2].weight, error(ps) end -svd_test(A, 3) \ No newline at end of file +for retraction in (Geodesic(), Cayley()) + svd_test(A, 3, retraction=retraction) +end \ No newline at end of file From 5ee8f5d0582beb5f9140e397648cf4c1d4ddff60 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 15:55:13 +0200 Subject: [PATCH 18/41] Moved file to src. --- docs/src/tutorials/mnist_tutorial.md | 59 ++++++++++++++++++++++++++++ 1 file changed, 59 insertions(+) create mode 100644 docs/src/tutorials/mnist_tutorial.md diff --git a/docs/src/tutorials/mnist_tutorial.md b/docs/src/tutorials/mnist_tutorial.md new file mode 100644 index 000000000..0527f7cce --- /dev/null +++ b/docs/src/tutorials/mnist_tutorial.md @@ -0,0 +1,59 @@ +# MNIST tutorial + +This is a short tutorial that shows how we can use `GeometricMachineLearning` to build a vision transformer and apply it for MNIST, while also putting some of the weights on a manifold. + +First, we need to import the relevant packages: + +```julia +using GeometricMachineLearning, CUDA +import Zygote, MLDatasets +``` + +In this example `Zygote` as an AD routine and we get the dataset from `MLDatasets`. First we need to load the data set, and put it on GPU (if you have one): + +```julia +train_x, train_y = MLDatasets.MNIST(split=:train)[:] +test_x, test_y = MLDatasets.MNIST(split=:test)[:] +train_x = train_x |> cu +test_x = test_x |> cu +train_y = train_y |> cu +test_y = test_y |> cu +``` + +`GeometricMachineLearning` has built-in data loaders that make it particularly easy to handle data: + +```julia +patch_length = 7 +dl = DataLoader(train_x, train_y, batch_size=512, patch_length=patch_length) +dl_test = DataLoader(train_x, train_y, batch_size=length(y), patch_length=patch_length) +``` + +The second line in the above code snippet indicates that we use the entire data set as one "batch" when processing the test set. For training, the batch size was here set to 512. + +```julia +ps = initialparameters(backend, eltype(dl.data), Ψᵉ) + +optimizer_instance = Optimizer(o, ps) + +println("initial test accuracy: ", accuracy(Ψᵉ, ps, dl_test), "\n") + +progress_object = Progress(n_training_steps; enabled=true) + +loss_array = zeros(eltype(train_x), n_training_steps) +for i in 1:n_training_steps + redraw_batch!(dl) + # get rid of try catch statement. This softmax issue should be solvable! + loss_val, pb = try Zygote.pullback(ps -> loss(Ψᵉ, ps, dl), ps) + catch + loss_array[i] = loss_array[i-1] + continue + end + dp = pb(one(loss_val))[1] + + optimization_step!(optimizer_instance, Ψᵉ, ps, dp) + ProgressMeter.next!(progress_object; showvalues = [(:TrainingLoss, loss_val)]) + loss_array[i] = loss_val +end + +println("final test accuracy: ", accuracy(Ψᵉ, ps, dl_test), "\n") +``` \ No newline at end of file From 81228e78d8267621ca71041db679afae6d310aa1 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 15:55:47 +0200 Subject: [PATCH 19/41] Changed path to tutorial. --- docs/make.jl | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/docs/make.jl b/docs/make.jl index 053216766..8ec4ddb1d 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -42,7 +42,7 @@ makedocs(; "Routines" => "data_loader/data_loader.md", ], "Tutorials" =>[ - "MNIST" => "../tutorials/mnist_tutorial.md", + "MNIST" => "tutorials/mnist_tutorial.md", ], "Library" => "library.md", ], From 596b4f5eb99031ef969000c7b629a1faaf3106ac Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 16:32:01 +0200 Subject: [PATCH 20/41] Moved directory. --- docs/tutorials/mnist_tutorial.md | 59 -------------------------------- 1 file changed, 59 deletions(-) delete mode 100644 docs/tutorials/mnist_tutorial.md diff --git a/docs/tutorials/mnist_tutorial.md b/docs/tutorials/mnist_tutorial.md deleted file mode 100644 index 0527f7cce..000000000 --- a/docs/tutorials/mnist_tutorial.md +++ /dev/null @@ -1,59 +0,0 @@ -# MNIST tutorial - -This is a short tutorial that shows how we can use `GeometricMachineLearning` to build a vision transformer and apply it for MNIST, while also putting some of the weights on a manifold. - -First, we need to import the relevant packages: - -```julia -using GeometricMachineLearning, CUDA -import Zygote, MLDatasets -``` - -In this example `Zygote` as an AD routine and we get the dataset from `MLDatasets`. First we need to load the data set, and put it on GPU (if you have one): - -```julia -train_x, train_y = MLDatasets.MNIST(split=:train)[:] -test_x, test_y = MLDatasets.MNIST(split=:test)[:] -train_x = train_x |> cu -test_x = test_x |> cu -train_y = train_y |> cu -test_y = test_y |> cu -``` - -`GeometricMachineLearning` has built-in data loaders that make it particularly easy to handle data: - -```julia -patch_length = 7 -dl = DataLoader(train_x, train_y, batch_size=512, patch_length=patch_length) -dl_test = DataLoader(train_x, train_y, batch_size=length(y), patch_length=patch_length) -``` - -The second line in the above code snippet indicates that we use the entire data set as one "batch" when processing the test set. For training, the batch size was here set to 512. - -```julia -ps = initialparameters(backend, eltype(dl.data), Ψᵉ) - -optimizer_instance = Optimizer(o, ps) - -println("initial test accuracy: ", accuracy(Ψᵉ, ps, dl_test), "\n") - -progress_object = Progress(n_training_steps; enabled=true) - -loss_array = zeros(eltype(train_x), n_training_steps) -for i in 1:n_training_steps - redraw_batch!(dl) - # get rid of try catch statement. This softmax issue should be solvable! - loss_val, pb = try Zygote.pullback(ps -> loss(Ψᵉ, ps, dl), ps) - catch - loss_array[i] = loss_array[i-1] - continue - end - dp = pb(one(loss_val))[1] - - optimization_step!(optimizer_instance, Ψᵉ, ps, dp) - ProgressMeter.next!(progress_object; showvalues = [(:TrainingLoss, loss_val)]) - loss_array[i] = loss_val -end - -println("final test accuracy: ", accuracy(Ψᵉ, ps, dl_test), "\n") -``` \ No newline at end of file From 337d3a432a59ddcd799cf73108033f947bbf138a Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 16:32:35 +0200 Subject: [PATCH 21/41] Fixed problem with Cayley transform. --- .../src/optimizers/manifold_related/cayley.md | 26 ++++++++++++------- .../manifold_related/retractions.jl | 7 ++--- 2 files changed, 21 insertions(+), 12 deletions(-) diff --git a/docs/src/optimizers/manifold_related/cayley.md b/docs/src/optimizers/manifold_related/cayley.md index febbc4ce2..15d814fc7 100644 --- a/docs/src/optimizers/manifold_related/cayley.md +++ b/docs/src/optimizers/manifold_related/cayley.md @@ -19,26 +19,34 @@ C = \begin{bmatrix} \end{bmatrix} = \begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} \begin{bmatrix} \mathbb{I} & \mathbb{O} \\ \frac{1}{2}A & -B^T \end{bmatrix}, ``` -where the second expression exploits the sparse structure of the array, i.e. it is a multiplication of a $N\times2n$ with a $2n\times{}N$ matrix. We can hence use the **Sherman-Morrison-Woodbury formula**: +where the second expression exploits the sparse structure of the array, i.e. it is a multiplication of a $N\times2n$ with a $2n\times{}N$ matrix. We can hence use the **Sherman-Morrison-Woodbury formula** to obtain: ```math -(\mathbb{I} - UV)^{-1} = \mathbb{I} + U(\mathbb{I} - VU)^{-1}V +(\mathbb{I} - \frac{1}{2}UV)^{-1} = \mathbb{I} + \frac{1}{2}U(\mathbb{I} - \frac{1}{2}VU)^{-1}V ``` -So What we have to invert is the term +So what we have to invert is the term ```math -\mathbb{I} - \begin{bmatrix} \mathbb{I} & \mathbb{O} \\ \frac{1}{2}A & -B^T \end{bmatrix}\begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} = -\begin{bmatrix} \mathbb{I} - \frac{1}{2}A & - \mathbb{I} \\ B^TB - \frac{1}{4}A^2 & \mathbb{I} - \frac{1}{2}A \end{bmatrix}. +\mathbb{I} - \frac{1}{2}\begin{bmatrix} \mathbb{I} & \mathbb{O} \\ \frac{1}{2}A & -B^T \end{bmatrix}\begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} = +\begin{bmatrix} \mathbb{I} - \frac{1}{4}A & - \frac{1}{2}\mathbb{I} \\ \frac{1}{2}B^TB - \frac{1}{8}A^2 & \mathbb{I} - \frac{1}{4}A \end{bmatrix}. ``` The whole cayley transform is then: -$$ -\begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} \begin{bmatrix} \mathbb{I} - \frac{1}{2}A & - \mathbb{I} \\ B^TB - \frac{1}{4}A^2 & \mathbb{I} - \frac{1}{2}A \end{bmatrix}^{-1} \begin{bmatrix} \mathbb{I} & \mathbb{O} \\ \frac{1}{2}A & -B^T \end{bmatrix}\left( E + \begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} \begin{bmatrix} \mathbb{I} \\ \frac{1}{2}A \end{bmatrix}\ \right) = \\ +```math +\left(\mathbb{I} + \frac{1}{2}\begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} \begin{bmatrix} \mathbb{I} - \frac{1}{4}A & - \frac{1}{2}\mathbb{I} \\ \frac{1}{2}B^TB - \frac{1}{8}A^2 & \mathbb{I} - \frac{1}{4}A \end{bmatrix}^{-1} \begin{bmatrix} \mathbb{I} & \mathbb{O} \\ \frac{1}{2}A & -B^T \end{bmatrix} \right)\left( E + \frac{1}{2}\begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} \begin{bmatrix} \mathbb{I} \\ \frac{1}{2}A \end{bmatrix}\ \right) = \\ + +E + \frac{1}{2}\begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix}\left( + \begin{bmatrix} \mathbb{I} \\ \frac{1}{2}A \end{bmatrix} + + \begin{bmatrix} \mathbb{I} - \frac{1}{4}A & - \frac{1}{2}\mathbb{I} \\ \frac{1}{2}B^TB - \frac{1}{8}A^2 & \mathbb{I} - \frac{1}{4}A \end{bmatrix}^{-1}\left( -\begin{bmatrix} \frac{1}{2}A & \mathbb{I} \\ B & \mathbb{O} \end{bmatrix} \begin{bmatrix} \mathbb{I} - \frac{1}{2}A & - \mathbb{I} \\ B^TB - \frac{1}{4}A^2 & \mathbb{I} - \frac{1}{2}A \end{bmatrix}^{-1}\left( \begin{bmatrix} \mathbb{I} \\ \frac{1}{2}A \end{bmatrix} + \begin{bmatrix} A \\ \frac{1}{2}A^2 - B^TB \end{bmatrix} \right). -$$ + \begin{bmatrix} \mathbb{I} \\ \frac{1}{2}A \end{bmatrix} + + \begin{bmatrix} \frac{1}{2}A \\ \frac{1}{4}A^2 - \frac{1}{2}B^TB \end{bmatrix} + + \right) + \right) +``` Note that for computational reason we compute $\mathrm{Cayley}(C)E$ instead of just the Cayley transform (see the section on [retractions](retractions.md)). \ No newline at end of file diff --git a/src/optimizers/manifold_related/retractions.jl b/src/optimizers/manifold_related/retractions.jl index 60cac6e0d..11b552406 100644 --- a/src/optimizers/manifold_related/retractions.jl +++ b/src/optimizers/manifold_related/retractions.jl @@ -75,11 +75,12 @@ function cayley(B::StiefelLieAlgHorMatrix{T}) where T A_mat2 = B.A*B.A BB = B.B'*B.B - exponent = hcat(vcat(unit - T(.5)*A_mat, BB - T(.25)*A_mat2), vcat(-unit, unit - T(.5)*A_mat)) + exponent = hcat(vcat(unit - T(.25)*A_mat, T(.5)*BB - T(.125)*A_mat2), vcat(-T(.5)*unit, unit - T(.25)*A_mat)) StiefelManifold( - hcat(vcat(T(.5)*A_mat, B.B), vcat(unit, zero(B.B)))* + E + + T(.5)*hcat(vcat(T(.5)*A_mat, B.B), vcat(unit, zero(B.B)))* ( - exponent \ (vcat(unit, T(0.5)*A_mat) + vcat(A_mat, T(0.5)*A_mat2 - BB)) + vcat(unit, T(0.5)*A_mat) + exponent \ (vcat(unit, T(0.5)*A_mat) + vcat(T(0.5)*A_mat, T(0.25)*A_mat2 - T(0.5)*BB)) ) ) end \ No newline at end of file From 4d6bdbcdd02fedb16d0e5fb93cc3be785457f5c0 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Wed, 13 Sep 2023 16:33:20 +0200 Subject: [PATCH 22/41] Changed argument input name to lower case retraction. --- src/layers/psd_like_layer.jl | 7 +++---- src/layers/stiefel_layer.jl | 6 +++--- 2 files changed, 6 insertions(+), 7 deletions(-) diff --git a/src/layers/psd_like_layer.jl b/src/layers/psd_like_layer.jl index 245ad5dd7..b3f10efd4 100644 --- a/src/layers/psd_like_layer.jl +++ b/src/layers/psd_like_layer.jl @@ -5,14 +5,13 @@ One layer has the following shape: |Φ 0| A = |0 Φ|, where Φ is an element of the regular Stiefel manifold. """ - -struct PSDLayer{M, N, retraction} <: LayerWithManifold{M, N, retraction} end +struct PSDLayer{M, N, Retraction} <: LayerWithManifold{M, N, Retraction} end default_retr = Geodesic() -function PSDLayer(M::Integer, N::Integer; Retraction=default_retr) +function PSDLayer(M::Integer, N::Integer; retraction=default_retr) @assert iseven(M) @assert iseven(N) - PSDLayer{M, N, typeof(Retraction)}() + PSDLayer{M, N, typeof(retraction)}() end function parameterlength(::PSDLayer{M, N}) where {M, N} diff --git a/src/layers/stiefel_layer.jl b/src/layers/stiefel_layer.jl index 23027aa8e..58c6839bd 100644 --- a/src/layers/stiefel_layer.jl +++ b/src/layers/stiefel_layer.jl @@ -1,11 +1,11 @@ """ Defines a layer that performs simple multiplication with an element of the Stiefel manifold. """ -struct StiefelLayer{M, N, retraction} <: ManifoldLayer{M, N, retraction} end +struct StiefelLayer{M, N, Retraction} <: ManifoldLayer{M, N, Retraction} end default_retr = Geodesic() -function StiefelLayer(n::Integer, N::Integer, Retraction::AbstractRetraction=default_retr) - StiefelLayer{n, N, typeof(Retraction)}() +function StiefelLayer(n::Integer, N::Integer; retraction::AbstractRetraction=default_retr) + StiefelLayer{n, N, typeof(retraction)}() end function AbstractNeuralNetworks.initialparameters(backend::KernelAbstractions.Backend, ::Type{T}, d::StiefelLayer{M,N}; rng::AbstractRNG=Random.default_rng()) where {M,N,T} From 6c1f3d90128dd16e41a29ed6378f3391da9212f6 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Thu, 14 Sep 2023 13:56:10 +0200 Subject: [PATCH 23/41] Added a script to generate data as in the weakly symplectic paper. --- .../symplectic_autoencoders/generate_data.jl | 64 +++++++++++++++++++ 1 file changed, 64 insertions(+) create mode 100644 scripts/symplectic_autoencoders/generate_data.jl diff --git a/scripts/symplectic_autoencoders/generate_data.jl b/scripts/symplectic_autoencoders/generate_data.jl new file mode 100644 index 000000000..8301e061a --- /dev/null +++ b/scripts/symplectic_autoencoders/generate_data.jl @@ -0,0 +1,64 @@ +using Plots + +function h(x::T) where T + if T(0) ≤ x ≤ T(1) + T(1) - T(1.5)*x^2 + T(.75)*x^3 + elseif T(1) < x ≤ 2 + T(.25)*(T(2) - x)^3 + else + T(0) + end +end + +function s(ξ::T, μ::T) where T + T(4) / μ * abs(ξ + T(.5)*(T(1) - μ)) +end + +u₀(ξ, μ) = h∘s(ξ, μ) +u(t, ξ, μ) = u₀(ξ .- μ * t, μ) + +function s(ξ::AbstractVector{T}, μ::T) where T + s_closure(ξ_scal) = s(ξ_scal, μ) + s_closure.(ξ) +end + +function u₀(ξ::AbstractVector{T}, μ::T) where T + h.(s(ξ, μ)) +end + +function get_domain(T=Float32, spacing=T(.01), time_step=T(0.01)) + Ω = T(-.5):spacing:T(.5) + I = T(0):time_step:T(1) + Ω, I +end + +function plot_time_evolution(T=Float32; spacing=T(.01), time_step=T(0.25), μ=T(.3)) + Ω, I = get_domain(T, spacing, time_step) + curves = zeros(T, length(Ω), length(I)) + + for it in zip(axes(I, 1), I) + i = it[1]; t = it[2] + curves[:, i] = u(t, Ω, μ) + end + curves, plot(Ω, curves, layout=(length(I), 1)) +end + +#= +μ = Float32(5/12) +data, p = plot_time_evolution(;μ=μ) +png(p, "data_for_μ="*string(μ)) +=# + +function generate_data(T=Float32; spacing=T(.01), time_step=T(0.01), μ_collection=T(5/12):T(.1):T(5/6)) + Ω, I = get_domain(T, spacing, time_step) + curves = zeros(T, length(Ω), length(μ_collection), length(I)) + + for it in zip(axes(I, 1), I) + i = it[1]; t = it[2] + for it2 in zip(axes(μ_collection, 1), μ_collection) + j = it2[1]; μ = it2[2] + curves[:, j, i] = u(t, Ω, μ) + end + end + curves +end \ No newline at end of file From 2c4a71b2f360c3a8e285d461770123655a957118 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Thu, 14 Sep 2023 13:56:44 +0200 Subject: [PATCH 24/41] Added routine for assigning a batch in the amtrix case. --- src/data_loader/matrix_assign.jl | 9 +++++++++ 1 file changed, 9 insertions(+) create mode 100644 src/data_loader/matrix_assign.jl diff --git a/src/data_loader/matrix_assign.jl b/src/data_loader/matrix_assign.jl new file mode 100644 index 000000000..f6ef93543 --- /dev/null +++ b/src/data_loader/matrix_assign.jl @@ -0,0 +1,9 @@ +""" +This assigns the batch if the data are in form of a matrix. +""" +function draw_batch!(batch::AbstractMatrix{T}, data::AbstractMatrix{T}) where T + sys_dim, batch_size = size(batch) + n_params = size(data, 2) + param_indices = Int.(ceil.(rand(T, batch_size)*n_params)) + batch .= data[:, param_indices] +end \ No newline at end of file From b3a4d1d1ca1982f1d9bab1adb71b78be87cd2586 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Thu, 14 Sep 2023 13:57:08 +0200 Subject: [PATCH 25/41] Added assign_tensor file. --- src/GeometricMachineLearning.jl | 1 + 1 file changed, 1 insertion(+) diff --git a/src/GeometricMachineLearning.jl b/src/GeometricMachineLearning.jl index 503bcbbee..938b5ff1c 100644 --- a/src/GeometricMachineLearning.jl +++ b/src/GeometricMachineLearning.jl @@ -68,6 +68,7 @@ module GeometricMachineLearning # export tensor_mat_mul include("data_loader/tensor_assign.jl") + include("data_loader/matrix_assign.jl") include("data_loader/data_loader.jl") include("data_loader/mnist_utils.jl") export DataLoader, redraw_batch!, loss, onehotbatch, accuracy From b84b8ba16e65d5676bbd0b19c4e850ab7785d9fc Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Thu, 14 Sep 2023 13:57:22 +0200 Subject: [PATCH 26/41] Added new constructor for matrix input. --- src/data_loader/data_loader.jl | 59 ++++++++++++++++++++-------------- 1 file changed, 34 insertions(+), 25 deletions(-) diff --git a/src/data_loader/data_loader.jl b/src/data_loader/data_loader.jl index e67005418..887113ae2 100644 --- a/src/data_loader/data_loader.jl +++ b/src/data_loader/data_loader.jl @@ -1,39 +1,39 @@ """ -Data Loader is a struct that creates an instance based on a tensor (or different input format) and is designed to make training convenient. - -Implemented: -If the data loader is called with a single tensor, a batch_size and an output_size, then the batch is drawn randomly in the relevant range and the output is assigned accordingly. - -The fields of the struct are the following: - - data: The input data with axes (i) system dimension, (ii) number of parameters and (iii) number of time steps. - - batch: A tensor in which the current batch is stored. - - target_tensor: A tensor in which the current target is stored. - - output: The tensor from which the output is drawn - this may be of type Nothing if the constructor is only called with one tensor. - - sys_dim: The ``dimension'' of the system, i.e. what is taken as input by a regular neural network. - - seq_length: The length batches should have. - - batch_size: - - output_size: The size of the second axis of the output tensor (prediction_window, output_size=1 in most cases) - - n_params: The number of parameters that are present in the data set (length of second axis). - - n_time_steps: Number of time steps (length of third axis). - -For drawing the batch, the sampling is done over n_params and n_time_steps (here seq_length and output_size are also taken into account). - -TODO: Implement DataLoader that works well with GeometricEnsembles etc. +Data Loader is a struct that creates an instance based on a tensor (or different input format) and is designed to make training convenient. \n +\n +Implemented: \n +If the data loader is called with a single tensor, a batch_size and an output_size, then the batch is drawn randomly in the relevant range and the output is assigned accordingly.\n +\n +The fields of the struct are the following: \n +\t - data: The input data with axes (i) system dimension, (ii) number of parameters and (iii) number of time steps.\n +\t - batch: A tensor in which the current batch is stored.\n +\t - target_tensor: A tensor in which the current target is stored.\n +\t - output: The tensor from which the output is drawn - this may be of type Nothing if the constructor is only called with one tensor.\n +\t - sys_dim: The ``dimension'' of the system, i.e. what is taken as input by a regular neural network.\n +\t - seq_length: The length batches should have.\n +\t - batch_size:\n +\t - output_size: The size of the second axis of the output tensor (prediction_window, output_size=1 in most cases)\n +\t - n_params: The number of parameters that are present in the data set (length of second axis)\n +\t - n_time_steps: Number of time steps (length of third axis)\n +\n +For drawing the batch, the sampling is done over n_params and n_time_steps (here seq_length and output_size are also taken into account).\n +\n +TODO: Implement DataLoader that works well with GeometricEnsembles etc.\n """ -struct DataLoader{T, AT<:AbstractArray{T}, TT<:Union{AbstractArray, Nothing}, OT<:AbstractArray} +struct DataLoader{T, AT<:AbstractArray{T}, TT<:Union{AbstractArray, Nothing}, OT<:Union{AbstractArray, Nothing}} data::AT batch::AT target_tensor::TT output::OT sys_dim::Integer - seq_length::Integer + seq_length::Union{Integer, Nothing} batch_size::Integer - output_size::Integer + output_size::Union{Integer, Nothing} n_params::Integer - n_time_steps::Integer + n_time_steps::Union{Integer, Nothing} end -function DataLoader(data::AbstractArray{T, 3}, seq_length=10, batch_size=32, output_size=1) where T +function DataLoader(data::AbstractArray{T, 3}; seq_length=10, batch_size=32, output_size=1) where T @info "You have provided a tensor with three axes as input. They will be interpreted as \n (i) system dimension, (ii) number of parameters and (iii) number of time steps." sys_dim,n_params,n_time_steps = size(data) backend = KernelAbstractions.get_backend(data) @@ -43,6 +43,15 @@ function DataLoader(data::AbstractArray{T, 3}, seq_length=10, batch_size=32, out DataLoader{T, typeof(data), Nothing, typeof(output)}(data, batch, nothing, output, sys_dim, seq_length, batch_size, output_size, n_params, n_time_steps) end +function DataLoader(data::AbstractMatrix{T}; batch_size=32) where T + @info "You have provided a matrix as input. The axes will be interpreted as (i) system dimension and (ii) number of parameters." + sys_dim, n_params = size(data) + backend = KernelAbstractions.get_backend(data) + batch = KernelAbstractions.allocate(backend, T, sys_dim, batch_size) + draw_batch!(batch, data) + DataLoader{T, typeof(data), Nothing, Nothing}(data, batch, nothing, nothing, sys_dim, nothing, batch_size, nothing, n_params, nothing) +end + # T and T1 are not the same because T1 is of Integer type function DataLoader(data::AbstractArray{T, 3}, target::AbstractVector{T1}; batch_size=32, patch_length=7) where {T, T1} @info "You provided a tensor and a vector as input. This will be treated as a classification problem (MNIST). Tensor axes: (i) & (ii) image axes and (iii) batch dimesnion." From 2f43c3923e59b374119cce40501b3bb77f7bb391 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Fri, 15 Sep 2023 12:10:53 +0200 Subject: [PATCH 27/41] Added new data loader functionality for autoencoders. --- src/data_loader/data_loader.jl | 9 +++++++++ 1 file changed, 9 insertions(+) diff --git a/src/data_loader/data_loader.jl b/src/data_loader/data_loader.jl index 887113ae2..d8f4bfe3c 100644 --- a/src/data_loader/data_loader.jl +++ b/src/data_loader/data_loader.jl @@ -77,12 +77,21 @@ function redraw_batch!(dl::DataLoader{T, AT, Nothing}) where {T, AT<:AbstractArr draw_batch!(dl.batch, dl.output, dl.data) end +function redraw_batch!(dl::DataLoader{T, AT, Nothing}) where {T, AT<:AbstractMatrix{T}} + draw_batch!(dl.batch, dl.data) +end + function loss(model::Union{Chain, AbstractExplicitLayer}, ps::Union{Tuple, NamedTuple}, dl::DataLoader{T}) where T batch_output = model(dl.batch, ps) output_estimate = assign_output_estimate(batch_output, dl.output_size) norm(dl.output - output_estimate)/T(sqrt(dl.batch_size))/T(sqrt(dl.output_size)) end +function loss(model::Chain, ps::Tuple, dl::DataLoader{T}) where T + batch_output = model(dl.batch, ps) + norm(batch_output - dl.batch)/norm(dl.batch) +end + function accuracy(model::Chain, ps::Tuple, dl::DataLoader{T, AT, BT}) where {T, T2<:Integer, AT<:AbstractArray{T}, BT<:AbstractArray{T2}} output_tensor = model(dl.batch, ps) output_estimate = assign_output_estimate(output_tensor, dl.output_size) From db76b115e6ff85d8305416e2cc57ef0942ddab08 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Fri, 15 Sep 2023 12:11:48 +0200 Subject: [PATCH 28/41] Fixed typos. Trainign data now correct. --- .../symplectic_autoencoders/generate_data.jl | 33 +++++++++++++------ 1 file changed, 23 insertions(+), 10 deletions(-) diff --git a/scripts/symplectic_autoencoders/generate_data.jl b/scripts/symplectic_autoencoders/generate_data.jl index 8301e061a..d8718a45a 100644 --- a/scripts/symplectic_autoencoders/generate_data.jl +++ b/scripts/symplectic_autoencoders/generate_data.jl @@ -1,4 +1,8 @@ -using Plots +""" +TODO: Implement p component! +""" + +using Plots, ForwardDiff function h(x::T) where T if T(0) ≤ x ≤ T(1) @@ -10,12 +14,17 @@ function h(x::T) where T end end -function s(ξ::T, μ::T) where T +function s(ξ, μ::T) where T T(4) / μ * abs(ξ + T(.5)*(T(1) - μ)) end -u₀(ξ, μ) = h∘s(ξ, μ) +u₀(ξ, μ) = h(s(ξ, μ)) u(t, ξ, μ) = u₀(ξ .- μ * t, μ) +p(t, ξ, μ) = ForwardDiff.derivative(t -> u(t,ξ,μ), t) +function p(t::T, ξ::AbstractVector{T}, μ::T) where T + p_closure(ξ) = p(t, ξ, μ) + p_closure.(ξ) +end function s(ξ::AbstractVector{T}, μ::T) where T s_closure(ξ_scal) = s(ξ_scal, μ) @@ -35,29 +44,33 @@ end function plot_time_evolution(T=Float32; spacing=T(.01), time_step=T(0.25), μ=T(.3)) Ω, I = get_domain(T, spacing, time_step) curves = zeros(T, length(Ω), length(I)) + curves_p = zeros(T, length(Ω), length(I)) for it in zip(axes(I, 1), I) i = it[1]; t = it[2] - curves[:, i] = u(t, Ω, μ) + curves[1:length(Ω), i] = u(t, Ω, μ) + curves_p[1:length(Ω), i] = p(t, Ω, μ) end - curves, plot(Ω, curves, layout=(length(I), 1)) + curves, curves_p, plot(Ω, curves, layout=(length(I), 1)), plot(Ω, curves_p, layout=(length(I), 1)) end -#= +#= μ = Float32(5/12) -data, p = plot_time_evolution(;μ=μ) -png(p, "data_for_μ="*string(μ)) +data, data_p, plot_q, plot_p = plot_time_evolution(;μ=μ) +png(plot_q, "q_data_for_μ="*string(μ)) +png(plot_p, "p_data_for_μ="*string(μ)) =# function generate_data(T=Float32; spacing=T(.01), time_step=T(0.01), μ_collection=T(5/12):T(.1):T(5/6)) Ω, I = get_domain(T, spacing, time_step) - curves = zeros(T, length(Ω), length(μ_collection), length(I)) + curves = zeros(T, 2*length(Ω), length(μ_collection), length(I)) for it in zip(axes(I, 1), I) i = it[1]; t = it[2] for it2 in zip(axes(μ_collection, 1), μ_collection) j = it2[1]; μ = it2[2] - curves[:, j, i] = u(t, Ω, μ) + curves[1:length(Ω), j, i] = u(t, Ω, μ) + curves[(length(Ω)+1):2*length(Ω), j, i] = p(t, Ω, μ) end end curves From e4487d79826a2cd830e1c1d82756ba1fa207d588 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Fri, 15 Sep 2023 12:12:27 +0200 Subject: [PATCH 29/41] Script for training symplectic autoencoder. --- scripts/symplectic_autoencoders/training.jl | 50 +++++++++++++++++++++ 1 file changed, 50 insertions(+) create mode 100644 scripts/symplectic_autoencoders/training.jl diff --git a/scripts/symplectic_autoencoders/training.jl b/scripts/symplectic_autoencoders/training.jl new file mode 100644 index 000000000..c4aeb3d8b --- /dev/null +++ b/scripts/symplectic_autoencoders/training.jl @@ -0,0 +1,50 @@ +using GeometricMachineLearning +using LinearAlgebra: svd, norm +using ProgressMeter +using Zygote +include("generate_data.jl") + +T = Float32 +spacing=T(.01) +time_step=T(0.01) +μ_collection=T(5/12):T(.1):T(5/6) +n = 5 +n_epochs = 2000 +backend = CPU() + +data = generate_data(T;spacing=spacing,time_step=time_step,μ_collection=μ_collection) +data = reshape(data, size(data,1), size(data,2)*size(data,3)) +N = size(data,1)÷2 +dl = DataLoader(data) +Φ = svd(hcat(data[1:N,:], data[(N+1):2*N,:])).U[:,1:n] +PSD = hcat(vcat(Φ, zero(Φ)), vcat(zero(Φ), Φ)) +PSD_error = norm(data - PSD*PSD'*data)/norm(data) + +activation = tanh +model = Chain( GradientQ(2*N, 2*N, activation), + GradientP(2*N, 2*N, activation), + PSDLayer(2*N, 2*n), + GradientQ(2*n, 2*n, activation), + GradientP(2*n, 2*n, activation), + GradientQ(2*n, 2*n, activation), + GradientP(2*n, 2*n, activation), + PSDLayer(2*n, 2*N), + GradientQ(2*N, 2*N, activation), + GradientP(2*N, 2*N, activation) +) + +ps = initialparameters(backend, Float32, model) +loss(model, ps, dl) + +optimizer_instance = Optimizer(AdamOptimizer(), ps) +n_training_iterations = Int(ceil(n_epochs*dl.n_params/dl.batch_size)) +progress_object = Progress(n_training_iterations; enabled=true) + +for _ in 1:n_training_iterations + redraw_batch!(dl) + loss_val, pb = Zygote.pullback(ps -> loss(model, ps, dl), ps) + dp = pb(one(loss_val))[1] + + optimization_step!(optimizer_instance, model, ps, dp) + ProgressMeter.next!(progress_object; showvalues=[(:TrainingLoss, loss_val)]) +end \ No newline at end of file From a4c77d2fdeb964febaee00c1a88b540cdd921533 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Sun, 17 Sep 2023 17:30:56 +0200 Subject: [PATCH 30/41] Started adding description(s) of autoencoder and symplectic autoencoder in ROM context. --- docs/src/tutorials/autoencoder.md | 45 ++++++++++++++++++++ docs/src/tutorials/symplectic_autoencoder.md | 12 ++++++ 2 files changed, 57 insertions(+) create mode 100644 docs/src/tutorials/autoencoder.md create mode 100644 docs/src/tutorials/symplectic_autoencoder.md diff --git a/docs/src/tutorials/autoencoder.md b/docs/src/tutorials/autoencoder.md new file mode 100644 index 000000000..177768f01 --- /dev/null +++ b/docs/src/tutorials/autoencoder.md @@ -0,0 +1,45 @@ +# Reduced Order Modelling and Autoencoders + +Reduced order modelling is a data-driven technique that exploits the structure of parametric PDEs to make solving those PDEs easier. + +Consider a parametric PDE written in the form: $F(z(\mu);\mu)=0$ where $z(\mu)$ evolves on a infinite-dimensional Hilbert space $V$. + +In modelling any PDE we have to choose a discretization (particle discretization, finite element method, ...) of V, which will be denoted by $V_h$. + +## Solution manifold + +To any parametric PDE we associate a **solution manifold**: + +```math +\mathcal{M} = \{z(\mu):F(z(\mu);\mu)=0, \mu\in\mathbb{P}\}. +``` + +![](images/solution_manifold_2.png) + +In the image above a 2-dimensional solution manifold is visualized as a sub-manifold in 3-dimensional space. In general the embedding space is an infinite-dimensional function space. + +As an example of this consider the 1-dimensional wave equation: + +```math +\partial_{tt}^2z(t,\xi;\mu) = \mu^2\partial_{\xi\xi}^2z(t,\xi;\mu)\text{ on }I\times\Omega, +``` +where $I = (0,1)$ and $\Omega=(-1/2,1/2)$. As initial condition for the first derivative we have $\partial_tz(0,\xi;\mu) = -\mu\partial_\xi{}z_0(\xi;\mu)$ and furthermore $z(t,\xi;\mu)=0$ on the boundary (i.e. $\xi\in\{-1/2,1/2\}$). + +The solution manifold is a 1-dimensional submanifold: + +```math +\mathcal{M} = \{(t, \xi)\mapsto{}u(t,\xi;\mu)=u_0(\xi-\mu{}t;\mu):\mu\in\mathbb{P}\subset\mathbb{R}\}. +``` + +If we provide an initial condition $u_0$, a parameter instance $\mu$ and a time $t$, then $\xi\mapsto{}z(t,\xi;\mu)$ will be the momentary solution. If we consider the time evolution of $z(t,\xi;\mu)$, then it evolves on a two-dimensional submanifold $\bar{\mathcal{M}} := \{\xi\mapsto{}z(t,\xi;\mu):t\in{}I\mu\in\mathbb{P}}$. + +## General workflow + +In reduced order modelling we aim to construct a mapping to a space that is close to this solution manifold. This is done through the following steps: + +1. Discretize the PDE. + +2. Solve the discretized PDE for a certain set of parameter instances $\mu\in\mathbb{P}$. + +3. Build a reduced basis with the data obtained from having solved the discretized PDE. This step consists of finding two mappings: the **reduction** $\mathcal{P}$ and the **reconstruction** $\mathcal{R}$. + diff --git a/docs/src/tutorials/symplectic_autoencoder.md b/docs/src/tutorials/symplectic_autoencoder.md new file mode 100644 index 000000000..81c89e749 --- /dev/null +++ b/docs/src/tutorials/symplectic_autoencoder.md @@ -0,0 +1,12 @@ +# Symplectic Autoencoder + +## Hamiltonian Model Order Reduction + +## Symplectic Solution Manifold + +In Hamiltonian reduced order modelling we try to find a symplectic submanifold of the solution space: $\tilde{\mathcal{M}} = \{\Psi^\mathrm{dec}(z_r)\in\mathbb{R}^{2N}:u_r\in\mathrm{R}^{2n}\}$ where $z_r$ is the reduced state of the system. + +The solution of the PDE always evolves on the solution manifold, and in case of a Hamiltonian PDE this evolution can be described through a Hamiltonian ODE. + +Theorem: Induction of Hamiltonian ODE on the solution manifold! +Multiplectic form!!! \ No newline at end of file From d9ec7f51e8c20fa56b13f96dd8ab3d2b0a78a010 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Mon, 18 Sep 2023 14:06:10 +0200 Subject: [PATCH 31/41] Moved files to new directory. --- docs/make.jl | 4 ++ .../src/reduced_order_modeling/autoencoder.md | 50 +++++++++++++++++++ .../symplectic_autoencoder.md | 45 +++++++++++++++++ docs/src/tutorials/symplectic_autoencoder.md | 12 ----- 4 files changed, 99 insertions(+), 12 deletions(-) create mode 100644 docs/src/reduced_order_modeling/autoencoder.md create mode 100644 docs/src/reduced_order_modeling/symplectic_autoencoder.md delete mode 100644 docs/src/tutorials/symplectic_autoencoder.md diff --git a/docs/make.jl b/docs/make.jl index 8ec4ddb1d..43032aba2 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -41,6 +41,10 @@ makedocs(; "Data Loader" =>[ "Routines" => "data_loader/data_loader.md", ], + "Reduced Order Modelling" =>[ + "POD and Autoencoders" => "reduced_order_modeling/autoencoder.md", + "PSD and Symplectic Autoencoders" => "reduced_order_modeling/symplectic_autoencoder.md", + ], "Tutorials" =>[ "MNIST" => "tutorials/mnist_tutorial.md", ], diff --git a/docs/src/reduced_order_modeling/autoencoder.md b/docs/src/reduced_order_modeling/autoencoder.md new file mode 100644 index 000000000..044cede0b --- /dev/null +++ b/docs/src/reduced_order_modeling/autoencoder.md @@ -0,0 +1,50 @@ +# Reduced Order modeling and Autoencoders + +Reduced order modeling is a data-driven technique that exploits the structure of parametric PDEs to make solving those PDEs easier. + +Consider a parametric PDE written in the form: $F(z(\mu);\mu)=0$ where $z(\mu)$ evolves on a infinite-dimensional Hilbert space $V$. + +In modeling any PDE we have to choose a discretization (particle discretization, finite element method, ...) of $V$, which will be denoted by $V_h$. + +## Solution manifold + +To any parametric PDE we associate a **solution manifold**: + +```math +\mathcal{M} = \{z(\mu):F(z(\mu);\mu)=0, \mu\in\mathbb{P}\}. +``` + +![](../images/solution_manifold_2.png) + +In the image above a 2-dimensional solution manifold is visualized as a sub-manifold in 3-dimensional space. In general the embedding space is an infinite-dimensional function space. + +As an example of this consider the 1-dimensional wave equation: + +```math +\partial_{tt}^2q(t,\xi;\mu) = \mu^2\partial_{\xi\xi}^2q(t,\xi;\mu)\text{ on }I\times\Omega, +``` +where $I = (0,1)$ and $\Omega=(-1/2,1/2)$. As initial condition for the first derivative we have $\partial_tq(0,\xi;\mu) = -\mu\partial_\xi{}q_0(\xi;\mu)$ and furthermore $q(t,\xi;\mu)=0$ on the boundary (i.e. $\xi\in\{-1/2,1/2\}$). + +The solution manifold is a 1-dimensional submanifold: + +```math +\mathcal{M} = \{(t, \xi)\mapsto{}q(t,\xi;\mu)=q_0(\xi-\mu{}t;\mu):\mu\in\mathbb{P}\subset\mathbb{R}\}. +``` + +If we provide an initial condition $u_0$, a parameter instance $\mu$ and a time $t$, then $\xi\mapsto{}q(t,\xi;\mu)$ will be the momentary solution. If we consider the time evolution of $q(t,\xi;\mu)$, then it evolves on a two-dimensional submanifold $\bar{\mathcal{M}} := \{\xi\mapsto{}q(t,\xi;\mu):t\in{}I,\mu\in\mathbb{P}\}$. + +## General workflow + +In reduced order modeling we aim to construct a mapping to a space that is close to this solution manifold. This is done through the following steps: + +1. Discretize the PDE. + +2. Solve the discretized PDE for a certain set of parameter instances $\mu\in\mathbb{P}$. + +3. Build a reduced basis with the data obtained from having solved the discretized PDE. This step consists of finding two mappings: the **reduction** $\mathcal{P}$ and the **reconstruction** $\mathcal{R}$. + +The third step can be done with various machine learning (ML) techniques. Traditionally the most popular of these has been **Proper orthogonal decomposition** (POD), but in recent years **autoencoders** have also become a popular alternative (see (Fresca et al, 2021)). + +## References + +- Fresca, Stefania, Luca Dede, and Andrea Manzoni. "A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs." Journal of Scientific Computing 87 (2021): 1-36. \ No newline at end of file diff --git a/docs/src/reduced_order_modeling/symplectic_autoencoder.md b/docs/src/reduced_order_modeling/symplectic_autoencoder.md new file mode 100644 index 000000000..fb8145ef4 --- /dev/null +++ b/docs/src/reduced_order_modeling/symplectic_autoencoder.md @@ -0,0 +1,45 @@ +# Symplectic Autoencoder + +Symplectic Autoencoders are a type of neural network suitable for treating Hamiltonian parametrized PDEs with slowly decaying Kolmogorov $n$-width. It is based on **proper symplectic decomposition** (PSD) and symplectic neural networks (SympNets). + +## Hamiltonian Model Order Reduction + +Hamiltonian PDEs are partial differential equations that, like its ODE counterpart, have a Hamiltonian associated with it. An example of this is the **linear wave equation** (see (Buckfink et al, 2023)) with Hamiltonian + +```math +\mathcal{H}(q, p; \mu) := \frac{1}{2}\int_\Omega\mu^2(\partial_\xi{}q(t,\xi;\mu))^2 + p(t,\xi;\mu)^2d\xi. +``` + +The PDE for to this Hamiltonian can be obtained similarly as in the ODE case: + +```math +\partial_t{}q(t,\xi;\mu) = \frac{\delta{}\mathcal{H}}{\delta{}p} = p(t,\xi;\mu), \quad \partial_t{}p(t,\xi;\mu) = -\frac{\delta{}\mathcal{H}}{\delta{}q} = \mu^2\partial_{\xi{}\xi}q(t,\xi;\mu) +``` + +## Symplectic Solution Manifold + +As with regular parametric PDEs, we also associate a solution manifold with Hamiltonian PDEs. This is a finite-dimensional manifold, on which the dynamics can be described through a Hamiltonian ODE. +I NEED A PROOF OR SOME EXPLANATION FOR THIS! + + +## Workflow for Symplectic ROM + +As with any other [reduced order modeling technique](autoencoder.md) we first discretize the PDE. This should be done with a structure-preserving scheme, thus yielding a (high-dimensional) Hamiltonian ODE as a result. Discretizing the wave equation above with finite differences yields a Hamiltonian system: + +```math +\mathcal{H}_\mathrm{discr}(z(t;\mu);\mu) := \frac{1}{2}x(t;\mu)^T\begin{bmatrix} -\mu^2D_{\xi{}\xi} & \mathbb{O} \\ \mathbb{O} & \mathbb{I} \end{bmatrix} x(t;\mu). +``` + +In Hamiltonian reduced order modelling we try to find a symplectic submanifold of the solution space[^1] that captures the dynamics of the full system as well as possible. + +[^1]: The submanifold is: $\tilde{\mathcal{M}} = \{\Psi^\mathrm{dec}(z_r)\in\mathbb{R}^{2N}:u_r\in\mathrm{R}^{2n}\}$ where $z_r$ is the reduced state of the system. + +Similar to the regular PDE case we again build an encoder $\Psi^\mathrm{enc}$ and a decoder $\Psi^\mathrm{dec}$; but now both these mappings are required to be symplectic! + +Concretely this means: +1. The encoder is a mapping from a high-dimensional symplectic space to a low-dimensional symplectic space, i.e. $\Psi^\mathrm{enc}:\mathbb{R}^{2N}\to\mathbb{R}^{2n}$ such that $\nabla\Psi^\mathrm{enc}\mathbb{J}_{2N}(\nabla\Psi^\mathrm{enc})^T = \mathbb{J}_{2n}$. +2. The decoder is a mapping from a low-dimensional symplectic space to a high-dimensional symplectic space, i.e. $\Psi^\mathrm{dec}:\mathbb{R}^{2n}\to\mathbb{R}^{2N}$ such that $(\nabla\Psi^\mathrm{dec})^T\mathbb{J}_{2N}\nabla\Psi^\mathrm{dec} = \mathbb{J}_{2n}$. + + +## References +- Buchfink, Patrick, Silke Glas, and Bernard Haasdonk. "Symplectic model reduction of Hamiltonian systems on nonlinear manifolds and approximation with weakly symplectic autoencoder." SIAM Journal on Scientific Computing 45.2 (2023): A289-A311. \ No newline at end of file diff --git a/docs/src/tutorials/symplectic_autoencoder.md b/docs/src/tutorials/symplectic_autoencoder.md deleted file mode 100644 index 81c89e749..000000000 --- a/docs/src/tutorials/symplectic_autoencoder.md +++ /dev/null @@ -1,12 +0,0 @@ -# Symplectic Autoencoder - -## Hamiltonian Model Order Reduction - -## Symplectic Solution Manifold - -In Hamiltonian reduced order modelling we try to find a symplectic submanifold of the solution space: $\tilde{\mathcal{M}} = \{\Psi^\mathrm{dec}(z_r)\in\mathbb{R}^{2N}:u_r\in\mathrm{R}^{2n}\}$ where $z_r$ is the reduced state of the system. - -The solution of the PDE always evolves on the solution manifold, and in case of a Hamiltonian PDE this evolution can be described through a Hamiltonian ODE. - -Theorem: Induction of Hamiltonian ODE on the solution manifold! -Multiplectic form!!! \ No newline at end of file From 232439297a8e9aef37cf405cbb4e74cf2df9db36 Mon 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z3xgI8quyKXH&ULAgBgoH6oymUr?7Mx2F1x~H-hp~&&x=(4)Tn=T7E}e17h18;Ekb2 znwgP-K@6@Ul>s&1i9J;ud%kupY3^Zh@jUoZtXsDpN5>d=H>M94(XS;aDmpqc@-0?W zPr73YDILN=#Ad~AV_k?y73NwSC?Da}gvVUSPo@9d-2CF@%Rl6fMOPae!D%{={?~SQ z90)K6d%ORh*W&yvhG9SxK`L&i=MqNm8X6e%bPxeOsY)43meuq~PUJF14XBZ(Q3&@smh~G6L@t0rEc7=%I zeLylHKH*gkFG15G`8@qL&WJykLl68(O_&n=Si;rdhg$@v0zdwEcl_b=6c3Fb`WPHz z{FLLP@rR-b+y;LB@0b5?!T#@%5I5ofE(VEuuuL!4jxmXbWt3PEO;v4`CrT%L|35SL BWIX@? literal 0 HcmV?d00001 From 23bf602649d10a8a48ab3b94d5ed6c61623f9aea Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Mon, 18 Sep 2023 14:07:16 +0200 Subject: [PATCH 33/41] Moved file. --- docs/src/tutorials/autoencoder.md | 45 ------------------------------- 1 file changed, 45 deletions(-) delete mode 100644 docs/src/tutorials/autoencoder.md diff --git a/docs/src/tutorials/autoencoder.md b/docs/src/tutorials/autoencoder.md deleted file mode 100644 index 177768f01..000000000 --- a/docs/src/tutorials/autoencoder.md +++ /dev/null @@ -1,45 +0,0 @@ -# Reduced Order Modelling and Autoencoders - -Reduced order modelling is a data-driven technique that exploits the structure of parametric PDEs to make solving those PDEs easier. - -Consider a parametric PDE written in the form: $F(z(\mu);\mu)=0$ where $z(\mu)$ evolves on a infinite-dimensional Hilbert space $V$. - -In modelling any PDE we have to choose a discretization (particle discretization, finite element method, ...) of V, which will be denoted by $V_h$. - -## Solution manifold - -To any parametric PDE we associate a **solution manifold**: - -```math -\mathcal{M} = \{z(\mu):F(z(\mu);\mu)=0, \mu\in\mathbb{P}\}. -``` - -![](images/solution_manifold_2.png) - -In the image above a 2-dimensional solution manifold is visualized as a sub-manifold in 3-dimensional space. In general the embedding space is an infinite-dimensional function space. - -As an example of this consider the 1-dimensional wave equation: - -```math -\partial_{tt}^2z(t,\xi;\mu) = \mu^2\partial_{\xi\xi}^2z(t,\xi;\mu)\text{ on }I\times\Omega, -``` -where $I = (0,1)$ and $\Omega=(-1/2,1/2)$. As initial condition for the first derivative we have $\partial_tz(0,\xi;\mu) = -\mu\partial_\xi{}z_0(\xi;\mu)$ and furthermore $z(t,\xi;\mu)=0$ on the boundary (i.e. $\xi\in\{-1/2,1/2\}$). - -The solution manifold is a 1-dimensional submanifold: - -```math -\mathcal{M} = \{(t, \xi)\mapsto{}u(t,\xi;\mu)=u_0(\xi-\mu{}t;\mu):\mu\in\mathbb{P}\subset\mathbb{R}\}. -``` - -If we provide an initial condition $u_0$, a parameter instance $\mu$ and a time $t$, then $\xi\mapsto{}z(t,\xi;\mu)$ will be the momentary solution. If we consider the time evolution of $z(t,\xi;\mu)$, then it evolves on a two-dimensional submanifold $\bar{\mathcal{M}} := \{\xi\mapsto{}z(t,\xi;\mu):t\in{}I\mu\in\mathbb{P}}$. - -## General workflow - -In reduced order modelling we aim to construct a mapping to a space that is close to this solution manifold. This is done through the following steps: - -1. Discretize the PDE. - -2. Solve the discretized PDE for a certain set of parameter instances $\mu\in\mathbb{P}$. - -3. Build a reduced basis with the data obtained from having solved the discretized PDE. This step consists of finding two mappings: the **reduction** $\mathcal{P}$ and the **reconstruction** $\mathcal{R}$. - From bdc9387c0928f32bca3a6576ad78d972b9e2cfa8 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Mon, 18 Sep 2023 14:07:59 +0200 Subject: [PATCH 34/41] Fixed test (forgot to divide a few routines with the sqrt factor before). --- src/layers/attention_layer.jl | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/src/layers/attention_layer.jl b/src/layers/attention_layer.jl index 7978813d6..8d5be53f5 100644 --- a/src/layers/attention_layer.jl +++ b/src/layers/attention_layer.jl @@ -63,14 +63,14 @@ function (d::Attention{M, M, Stiefel, Retraction, true})(x::AbstractMatrix{T}, p dim, input_length = size(x) @assert dim == M - x + x*d.activation((ps.PQ'*x)'*(ps.PK'*x)) + x + x*d.activation((ps.PQ'*x)'*(ps.PK'*x)/T(sqrt(M))) end function (d::Attention{M, M, Stiefel, Retraction, false})(x::AbstractMatrix{T}, ps::NamedTuple) where {M, Stiefel, Retraction, T} dim, input_length = size(x) @assert dim == M - x*d.activation((ps.PQ'*x)'*(ps.PK'*x)) + x*d.activation((ps.PQ'*x)'*(ps.PK'*x)/T(sqrt(M))) end function (d::Attention{M, M, Stiefel, Retraction, true})(x::AbstractArray{T, 3}, ps::NamedTuple) where {M, Stiefel, Retraction, T} @@ -80,7 +80,7 @@ function (d::Attention{M, M, Stiefel, Retraction, true})(x::AbstractArray{T, 3}, Q_tensor = mat_tensor_mul(ps.PQ', x) K_tensor = mat_tensor_mul(ps.PK', x) QK_tensor = tensor_transpose_tensor_mul(Q_tensor, K_tensor) - x + tensor_tensor_mul(x, d.activation(QK_tensor)) + x + tensor_tensor_mul(x, d.activation(QK_tensor/T(sqrt(M)))) end function (d::Attention{M, M, Stiefel, Retraction, false})(x::AbstractArray{T, 3}, ps::NamedTuple) where {M, Stiefel, Retraction, T} @@ -90,7 +90,7 @@ function (d::Attention{M, M, Stiefel, Retraction, false})(x::AbstractArray{T, 3} Q_tensor = mat_tensor_mul(ps.PQ', x) K_tensor = mat_tensor_mul(ps.PK', x) QK_tensor = tensor_transpose_tensor_mul(Q_tensor, K_tensor) - tensor_tensor_mul(x, d.activation(QK_tensor)) + tensor_tensor_mul(x, d.activation(QK_tensor/T(sqrt(M)))) end @kernel function upper_triangular_asymmetrize_kernel!(output::AbstractArray{T, 3}, input::AbstractArray{T, 3}) where T From b63d60ff6f09ed091467ce962def8f627946862b Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Mon, 18 Sep 2023 17:23:44 +0200 Subject: [PATCH 35/41] Started adding description to struct TrainingData because this was causing a warning. --- src/data/data_training.jl | 10 ++++++++++ 1 file changed, 10 insertions(+) diff --git a/src/data/data_training.jl b/src/data/data_training.jl index 6646d626b..215f0805b 100644 --- a/src/data/data_training.jl +++ b/src/data/data_training.jl @@ -1,5 +1,15 @@ abstract type AbstractTrainingData end + +""" +TrainingData stores: \n +\t - problem \n +\t - shape \n +\t - get \n +\t - symbols \n +\t - dim \n +\t - noisemaker \n +""" struct TrainingData{TK <: DataSymbol, TS <: AbstractDataShape, TP <: AbstractProblem, TG <: Dict{Symbol, <:Base.Callable}, TN <: Base.Callable} <: AbstractTrainingData problem::TP shape::TS From 07136aa6253ee803c5f6ce16fa8fb898c9bcf4a0 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Mon, 18 Sep 2023 17:24:47 +0200 Subject: [PATCH 36/41] Added kolmogorov n-width. --- docs/make.jl | 2 ++ 1 file changed, 2 insertions(+) diff --git a/docs/make.jl b/docs/make.jl index 43032aba2..0b9b97dcb 100644 --- a/docs/make.jl +++ b/docs/make.jl @@ -40,10 +40,12 @@ makedocs(; ], "Data Loader" =>[ "Routines" => "data_loader/data_loader.md", + "Snapshot matrix" => "data_loader/snapshot_matrix.md", ], "Reduced Order Modelling" =>[ "POD and Autoencoders" => "reduced_order_modeling/autoencoder.md", "PSD and Symplectic Autoencoders" => "reduced_order_modeling/symplectic_autoencoder.md", + "Kolmogorov n-width" => "reduced_order_modeling/kolmogorov_n_width.md", ], "Tutorials" =>[ "MNIST" => "tutorials/mnist_tutorial.md", From 6fe36b3cdabff57a5b07ee4a1f1f935fd386a19e Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Mon, 18 Sep 2023 17:25:13 +0200 Subject: [PATCH 37/41] Added PSD and started adding symplectic autoencoders. --- .../symplectic_autoencoder.md | 38 ++++++++++++++++++- 1 file changed, 37 insertions(+), 1 deletion(-) diff --git a/docs/src/reduced_order_modeling/symplectic_autoencoder.md b/docs/src/reduced_order_modeling/symplectic_autoencoder.md index fb8145ef4..d3b923fc2 100644 --- a/docs/src/reduced_order_modeling/symplectic_autoencoder.md +++ b/docs/src/reduced_order_modeling/symplectic_autoencoder.md @@ -40,6 +40,42 @@ Concretely this means: 1. The encoder is a mapping from a high-dimensional symplectic space to a low-dimensional symplectic space, i.e. $\Psi^\mathrm{enc}:\mathbb{R}^{2N}\to\mathbb{R}^{2n}$ such that $\nabla\Psi^\mathrm{enc}\mathbb{J}_{2N}(\nabla\Psi^\mathrm{enc})^T = \mathbb{J}_{2n}$. 2. The decoder is a mapping from a low-dimensional symplectic space to a high-dimensional symplectic space, i.e. $\Psi^\mathrm{dec}:\mathbb{R}^{2n}\to\mathbb{R}^{2N}$ such that $(\nabla\Psi^\mathrm{dec})^T\mathbb{J}_{2N}\nabla\Psi^\mathrm{dec} = \mathbb{J}_{2n}$. +If these two maps are constrained to linear maps, then one can easily find good solutions with **proper symplectic decomposition** (PSD). + +## Proper Symplectic Decomposition + +For PSD the two mappings $\Psi^\mathrm{enc}$ and $\Psi^\mathrm{dec}$ are constrained to be linear, orthonormal (i.e. $\Psi^T\Psi = \mathbb{I}$) and symplectic. The easiest way to enforce this is through the so-called **cotangent lift**: + +```math +\Psi_\mathrm{CL} = +\begin{bmatrix} \Phi & \mathbb{O} \\ \mathbb{O} & \Phi \end{bmatrix}, +``` +and $\Phi\in{}St(n,N)\sub\mathbb{R}^{N\times{}n}$, i.e. is an element of the [Stiefel manifold](../manifolds/stiefel_manifold.md). If the [snapshot matrix](../data_loader/snapshot_matrix.md) is of the form: + +```math +M = \left[\begin{array}{c:c:c:c} +\hat{q}_1(t_0) & \hat{q}_1(t_1) & \quad\ldots\quad & \hat{q}_1(t_f) \\ +\hat{q}_2(t_0) & \hat{q}_2(t_1) & \ldots & \hat{q}_2(t_f) \\ +\ldots & \ldots & \ldots & \ldots \\ +\hat{q}_N(t_0) & \hat{q}_N(t_1) & \ldots & \hat{q}_N(t_f) \\ +\hat{p}_1(t_0) & \hat{p}_1(t_1) & \ldots & \hat{p}_1(t_f) \\ +\hat{p}_2(t_0) & \hat{p}_2(t_1) & \ldots & \hat{p}_2(t_f) \\ +\ldots & \ldots & \ldots & \ldots \\ +\hat{p}_{N}(t_0) & \hat{p}_{N}(t_1) & \ldots & \hat{p}_{N}(t_f) \\ +\end{array}\right], +``` + +then $\Phi$ can be computed in a very straight-forward manner: +1. Rearrange the rows of the matrix $M$ such that we end up with a $N\times2(f+1)$ matrix: $\hat{M} := [M_q, M_p]$. +2. Perform SVD: $\hat{M} = U\Sigma{}V^T$; set $\Phi\gets{}U\mathtt{[:,1:n]}$. + +For details on the cotangent lift (and other methods for linear symplectic model reduction) consult (Peng and Mohseni, 2016). + +## Symplectic Autoencoders + +PSD suffers from the similar shortcomings as regular POD: it is a linear map and the approximation space $\tilde{\mathcal{M}}= \{\Psi^\mathrm{dec}(z_r)\in\mathbb{R}^{2N}:u_r\in\mathrm{R}^{2n}\}$ is strictly linear. For problems with slowly-decaying Kolmogorov $n$-width this leads to very poor approximations. + ## References -- Buchfink, Patrick, Silke Glas, and Bernard Haasdonk. "Symplectic model reduction of Hamiltonian systems on nonlinear manifolds and approximation with weakly symplectic autoencoder." SIAM Journal on Scientific Computing 45.2 (2023): A289-A311. \ No newline at end of file +- Buchfink, Patrick, Silke Glas, and Bernard Haasdonk. "Symplectic model reduction of Hamiltonian systems on nonlinear manifolds and approximation with weakly symplectic autoencoder." SIAM Journal on Scientific Computing 45.2 (2023): A289-A311. +- Peng, Liqian, and Kamran Mohseni. "Symplectic model reduction of Hamiltonian systems." SIAM Journal on Scientific Computing 38.1 (2016): A1-A27. \ No newline at end of file From 94108d18e5bbba17a8169b3401583d328272eb5c Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Mon, 18 Sep 2023 17:38:51 +0200 Subject: [PATCH 38/41] Added short description of Kolmogorov n-width. --- .../kolmogorov_n_width.md | 17 +++++++++++++++++ 1 file changed, 17 insertions(+) create mode 100644 docs/src/reduced_order_modeling/kolmogorov_n_width.md diff --git a/docs/src/reduced_order_modeling/kolmogorov_n_width.md b/docs/src/reduced_order_modeling/kolmogorov_n_width.md new file mode 100644 index 000000000..65595990b --- /dev/null +++ b/docs/src/reduced_order_modeling/kolmogorov_n_width.md @@ -0,0 +1,17 @@ +# Kolmogorov $n$-width + +The Kolmogorov $n$-width measures how well some set $\mathcal{M}$ (typically the solution manifold) can be approximated with a linear subspace: + +```math +d_n(\mathcal{M}) := \mathrm{inf}_{V_n\sub{}V;\mathrm{dim}V_n=n}\mathrm{sup}(u\in\mathcal{M})\mathrm{inf}_{v_n\in{}V_n}|| u - v_n ||_V, +``` + +with $\mathcal{M}\sub{}V$ and $V$ is a (typically infinite-dimensional) Banach space. For advection-dominated problems (among others) the **decay of the Kolmogorov $n$-width is very slow**, i.e. one has to pick $n$ very high in order to obtain useful approximations (see (Greif and Urban, 2019) and (Blickhan, 2023)). + +In order to overcome this, techniques based on neural networks (see e.g. (Lee and Carlberg, 2020)) and optimal transport (see e.g. (Blickhan, 2023)) have been used. + + +## References +- Greif, Constantin, and Karsten Urban. "Decay of the Kolmogorov N-width for wave problems." Applied Mathematics Letters 96 (2019): 216-222. +- Blickhan, Tobias. "A registration method for reduced basis problems using linear optimal transport." arXiv preprint arXiv:2304.14884 (2023). +- Lee, Kookjin, and Kevin T. Carlberg. "Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders." Journal of Computational Physics 404 (2020): 108973. From d0e1e610e72476facc596632aab6277f99c70392 Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Mon, 18 Sep 2023 17:39:07 +0200 Subject: [PATCH 39/41] Added short description of snapshot matrix. --- docs/src/data_loader/snapshot_matrix.md | 15 +++++++++++++++ 1 file changed, 15 insertions(+) create mode 100644 docs/src/data_loader/snapshot_matrix.md diff --git a/docs/src/data_loader/snapshot_matrix.md b/docs/src/data_loader/snapshot_matrix.md new file mode 100644 index 000000000..d2373aaf4 --- /dev/null +++ b/docs/src/data_loader/snapshot_matrix.md @@ -0,0 +1,15 @@ +# Snapshot matrix + +The snapshot matrix stores solutions of the high-dimensional ODE (obtained from discretizing a PDE). This is then used to construct [reduced bases](../reduced_order_modeling/autoencoder.md) in a data-driven way. So (for a single parameter[^1]) the snapshot matrix takes the following form: + +[^1]: If we deal with a parametrized PDE then there are **two stages** at which the snapshot matrix has to be processed: the offline stage and the online stage. + +```math +M = \left[\begin{array}{c:c:c:c} +\hat{u}_1(t_0) & \hat{u}_1(t_1) & \quad\ldots\quad & \hat{u}_1(t_f) \\ +\hat{u}_2(t_0) & \hat{u}_2(t_1) & \ldots & \hat{u}_2(t_f) \\ +\hat{u}_3(t_0) & \hat{u}_3(t_1) & \ldots & \hat{u}_3(t_f) \\ +\ldots & \ldots & \ldots & \ldots \\ +\hat{u}_{2N}(t_0) & \hat{u}_{2N}(t_1) & \ldots & \hat{u}_{2N}(t_f) \\ +\end{array}\right]. +``` \ No newline at end of file From 782778eb2ca54dd6c901333b0e5d69c24ddaab8f Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Mon, 18 Sep 2023 17:53:13 +0200 Subject: [PATCH 40/41] Images for symplectic autoencoder and mha. --- docs/src/images/mha.png | Bin 0 -> 73496 bytes docs/src/images/symplectic_autoencoder.png | Bin 0 -> 59941 bytes 2 files changed, 0 insertions(+), 0 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z*ec5H4791DD5G?*iHbDt5jbtFU_m`&0ZhIa6E|IH^Eij$7VxA;AeF6dD-mI7?YK0> zk{lYow`ktAZSwT=Zqt*M6n+VTQO9u_`x`2LPyut><(plNc|S^M4u#(BC_ zjAQin!k_FOf#LRpPqw3Mo1p7_HNNEpT@=t%tADX$`6N)d<)~giqc9_S-I-!*Z-47658}3b4~4~>Qmyx!yWL9nD+hG&GZLxSx)E@c})r8%VQKY zwdZ@SVU|%{JlE9C0MPMrG2E)i&73byyKT(Ye9gSVhM7F9N+>rjP6MiYBddJH43Fs7 z4J%UrJ*@=@EX~}5+;v~6q)k&$fj2WWt#eog9MsE6zg=Q+-|FhJ)ET+$EyKgoWP8U! z7>)Wf@8FQ5+{?~)xq>|tg-e;R z?4_EuHmWmp|ysAJswC4%H6d z_u-fH8-#k1zmLc$svSUW(csb3yMRu9`>*8mFY#CJl>A!O;XVQp;fp0JHAI>JDm MpFBcHJb(Lt0SL1L`Tzg` literal 0 HcmV?d00001 From 2f5185375c9822d04e8319c187418a308a4bc19c Mon Sep 17 00:00:00 2001 From: benedict-96 Date: Mon, 18 Sep 2023 17:56:38 +0200 Subject: [PATCH 41/41] Added image and some description for symplectic autoencoder. --- docs/src/reduced_order_modeling/symplectic_autoencoder.md | 8 +++++++- 1 file changed, 7 insertions(+), 1 deletion(-) diff --git a/docs/src/reduced_order_modeling/symplectic_autoencoder.md b/docs/src/reduced_order_modeling/symplectic_autoencoder.md index d3b923fc2..3d5d40a97 100644 --- a/docs/src/reduced_order_modeling/symplectic_autoencoder.md +++ b/docs/src/reduced_order_modeling/symplectic_autoencoder.md @@ -73,7 +73,13 @@ For details on the cotangent lift (and other methods for linear symplectic model ## Symplectic Autoencoders -PSD suffers from the similar shortcomings as regular POD: it is a linear map and the approximation space $\tilde{\mathcal{M}}= \{\Psi^\mathrm{dec}(z_r)\in\mathbb{R}^{2N}:u_r\in\mathrm{R}^{2n}\}$ is strictly linear. For problems with slowly-decaying Kolmogorov $n$-width this leads to very poor approximations. +PSD suffers from the similar shortcomings as regular POD: it is a linear map and the approximation space $\tilde{\mathcal{M}}= \{\Psi^\mathrm{dec}(z_r)\in\mathbb{R}^{2N}:u_r\in\mathrm{R}^{2n}\}$ is strictly linear. For problems with slowly-decaying [Kolmogorov $n$-width](kolmogorov_n_width.md) this leads to very poor approximations. + +In order to overcome this difficulty we use neural networks, more specifically [SympNets](../architectures/sympnet.md), together with cotangent lift-like matrices. The resulting architecture, symplectic autoencoders, are demonstrated in the following image: + +![](../images/symplectic_autoencoder.png) + +So we alternate between SympNet and PSD layers. Because all the PSD layers are based on matrices $\Phi\in{}St(n,N)$ we have to [optimize on the Stiefel manifold](../Optimizer.md). ## References