From 97ee1727a442740cd4c05e9861ac0bf931695dd0 Mon Sep 17 00:00:00 2001 From: lcnbr Date: Mon, 3 Jun 2024 12:03:25 +0200 Subject: [PATCH] docs: :memo: added presentation for FORM and Symbolica dev conf --- examples/gamma_chain.rs | 4 +- examples/gamma_network.rs | 2 +- examples/pres.rs | 138 + flake.nix | 2 + pres/diag.svg | 345 +++ pres/luLogo.png | Bin 0 -> 135138 bytes pres/pres.html | 5779 +++++++++++++++++++++++++++++++++++++ pres/pres.qmd | 738 +++++ pres/spenso.png | Bin 0 -> 10932 bytes src/structure.rs | 2 + src/tests.rs | 70 +- 11 files changed, 7069 insertions(+), 11 deletions(-) create mode 100644 examples/pres.rs create mode 100644 pres/diag.svg create mode 100644 pres/luLogo.png create mode 100644 pres/pres.html create mode 100644 pres/pres.qmd create mode 100644 pres/spenso.png diff --git a/examples/gamma_chain.rs b/examples/gamma_chain.rs index 072047f..177b8d0 100644 --- a/examples/gamma_chain.rs +++ b/examples/gamma_chain.rs @@ -227,7 +227,7 @@ fn main() { one.mul_fallible(4.3).unwrap(), ]; - let spacings: [i32; 2] = [20, 24]; + let spacings: [i32; 2] = [2, 4]; let mut start = 1; let mut ranges = Vec::new(); @@ -268,7 +268,7 @@ fn main() { #[cfg(feature = "shadowing")] { - let mut chain = gamma_net(&vec, vbar, u); + let mut chain = gamma_net_param(&vec); println!("{}", chain.graph.edges.len()); println!("{}", chain.graph.nodes.len()); println!("{}", chain.graph.involution.len()); diff --git a/examples/gamma_network.rs b/examples/gamma_network.rs index 55722f6..3cd2432 100644 --- a/examples/gamma_network.rs +++ b/examples/gamma_network.rs @@ -59,7 +59,7 @@ fn main() { one.mul_fallible(4.3).unwrap(), ]; - let spacings: [i32; 2] = [20, 24]; + let spacings: [i32; 2] = [2, 4]; let mut start = 1; let mut ranges = Vec::new(); diff --git a/examples/pres.rs b/examples/pres.rs new file mode 100644 index 0000000..c26f751 --- /dev/null +++ b/examples/pres.rs @@ -0,0 +1,138 @@ +use std::ops::Neg; + +use spenso::{ + ufo::{ + euclidean_four_vector, euclidean_four_vector_sym, gamma, gammasym, mink_four_vector, + mink_four_vector_sym, param_euclidean_four_vector, param_mink_four_vector, + }, + AbstractIndex, Complex, Contract, Dimension, FallibleMul, HasName, HasStructure, + HistoryStructure, MixedTensor, NamedStructure, NumTensor, Representation, Shadowable, Slot, + SparseTensor, SymbolicTensor, TensorNetwork, +}; + +use criterion::{criterion_group, criterion_main, Criterion}; +use num::ToPrimitive; + +use symbolica::atom::{Atom, Symbol}; + +fn gamma_net_sym( + minkindices: &[i32], + vbar: [Complex; 4], + u: [Complex; 4], +) -> TensorNetwork>> { + let mut i = 0; + let mut contracting_index = 0.into(); + let mut result: Vec>> = + vec![euclidean_four_vector_sym(contracting_index, &vbar).into()]; + for m in minkindices { + let ui = contracting_index; + contracting_index += 1.into(); + let uj = contracting_index; + if *m > 0 { + let p = [ + Complex::::new(1.0 + 0.01 * i.to_f64().unwrap(), 0.0), + Complex::::new(1.1 + 0.01 * i.to_f64().unwrap(), 0.0), + Complex::::new(1.2 + 0.01 * i.to_f64().unwrap(), 0.0), + Complex::::new(1.3 + 0.01 * i.to_f64().unwrap(), 0.0), + ]; + i += 1; + result.push(mink_four_vector_sym(usize::try_from(*m).unwrap().into(), &p).into()); + result.push(gammasym(usize::try_from(*m).unwrap().into(), (ui, uj)).into()); + } else { + result.push( + gammasym( + AbstractIndex::from(usize::try_from(m.neg()).unwrap() + 10000), + (ui, uj), + ) + .into(), + ); + } + } + result.push(euclidean_four_vector_sym(contracting_index, &u).into()); + TensorNetwork::from(result) +} + +fn indices(n: i32, m: i32) -> Vec { + let spacings: [i32; 2] = [n, m]; + let mut start = 1; + let mut ranges = Vec::new(); + + for &spacing in spacings.iter() { + ranges.push((start..start + spacing).chain(std::iter::once(-1))); + start += spacing; + } + + ranges.into_iter().flatten().collect() +} + +fn main() { + let one = Complex::::new(1.0, 0.0); + let _zero = Complex::::new(0.0, 0.0); + + let vbar = [ + one.mul_fallible(3.0).unwrap(), + one.mul_fallible(3.1).unwrap(), + one.mul_fallible(3.2).unwrap(), + one.mul_fallible(3.3).unwrap(), + ]; + let u = [ + one.mul_fallible(4.0).unwrap(), + one.mul_fallible(4.1).unwrap(), + one.mul_fallible(4.2).unwrap(), + one.mul_fallible(4.3).unwrap(), + ]; + let minkindices = indices(20, 24); + + let mut netsym = gamma_net_sym(&minkindices, vbar, u); + + println!("one:{}", netsym.dot()); + netsym.contract_algo(|s| TensorNetwork::edge_to_min_degree_node_with_depth(&s, 5)); + println!("two:{}", netsym.dot()); + netsym.contract_algo(|s| TensorNetwork::edge_to_min_degree_node_with_depth(&s, 10)); + println!("three:{}", netsym.dot()); + + let mink = Representation::Lorentz(Dimension(4)); + let mu = Slot::from((AbstractIndex(0), mink)); + let spin = Representation::SpinFundamental(Dimension(4)); + let spina = Representation::SpinAntiFundamental(Dimension(4)); + + let i = Slot::from((AbstractIndex(1), spin)); + let j = Slot::from((AbstractIndex(2), spina)); + let k = Slot::from((9.into(), spin)); + + let structure = NamedStructure::from_slots(vec![mu, i, j], "γ"); + let p_struct = NamedStructure::from_slots(vec![mu], "p"); + let t_struct = NamedStructure::from_slots(vec![i, j, k], "T"); + + let gamma_sym = SymbolicTensor::from_named(&structure).unwrap(); + let p_sym = SymbolicTensor::from_named(&p_struct).unwrap(); + let t_sym = SymbolicTensor::from_named(&t_struct).unwrap(); + + let f = gamma_sym + .contract(&p_sym) + .unwrap() + .contract(&t_sym) + .unwrap(); + + println!("{}", *f.get_atom()); + + let a: TensorNetwork = f.to_network().unwrap(); + + // println!("{}", a.dot()); + + // let γ1: MixedTensor<_> = gamma(1.into(), (1.into(), 2.into())).into(); + + // let γ2: MixedTensor<_> = gamma(10.into(), (2.into(), 3.into())).into(); + + let p1: MixedTensor<_> = param_mink_four_vector(1.into(), "p1").into(); + + let u: MixedTensor<_> = NamedStructure::new( + &[(1.into(), Representation::SpinFundamental(4.into()))], + "u", + ) + .shadow() + .unwrap() + .into(); + + println!("{}", u.to_symbolic().unwrap()); +} diff --git a/flake.nix b/flake.nix index df753d4..c9b32d6 100644 --- a/flake.nix +++ b/flake.nix @@ -174,6 +174,8 @@ pkgs.cargo-insta pkgs.git # pkgs.ripgrep + pkgs.quarto + pkgs.deno ]; }; }); diff --git a/pres/diag.svg b/pres/diag.svg new file mode 100644 index 0000000..6798467 --- /dev/null +++ b/pres/diag.svg @@ -0,0 +1,345 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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100644 index 0000000..3e235b6 --- /dev/null +++ b/pres/pres.qmd @@ -0,0 +1,738 @@ +--- +title: "Spenso" +subtitle: "a rust-based framework for symbolic and numerical tensor computation" +author: "Lucien Huber" +institute: "Bern University" +format: + revealjs: + slide-number: true + chalkboard: false + auto-animate: true + incremental: true + transition: slide + transition-speed: fast + preview-links: auto + logo: luLogo.png + code-overflow: wrap + highlight-style: breeze + embed-resources: true + footnotes-hover: true +--- + +# Introduction + +Tensors are pervasive in High Energy Physics. + + +We wanted a library that implements tensor contractions, and arithmetic, component wise and synergises well with symbolica. + + +::: {.notes} + For our project with Mathjis, Zeno, and Valentin we wanted a library that implements tensor contractions, and arithmetic, all component wise. + + I will go more in depth, into our use case but tensors have many uses. +::: + +## Uses + +Evaluating: + +- Feynman Rules (e.g. gamma chains) +- IBP (multi-dim. array) +- Tensor Networks for lattice QCD + +::: {.notes} + Component wise tensors are good for evaluating applied feynman rules of course. (lorentz and spinor structure) +::: + +## It already exists, no? + +Not really + +- Mathematica + - Too slow +- NDarray (Rust) + - Wanted Einstein/Abstract Index notation + - Needed Generic Dimensions +- cuTensorNet (CUDA) + - Support for non-euclidean metric + - Physical indices +- ITensors.jl (Julia/C++) + - Wanted Rust (for LU/gammaloop), zero-cost abstractions FTW + - Does not talk Symbolica + + +::: {.notes} + Why make such a library when this isn't a new concept? + + Well no package really fits our bill.. + + Mathematica: we need to evaluate tensor contractions at each integration point, it needs to be fast + + We use rust: for nice language features and speed. + + Rust: tensor package ecosystem is barren + + if we wanted speed: + + cutensoret + + there does exist a very close in feature set library +::: + + + +# Introducing Spenso! + +![](spenso.png){fig-align="center" width=100} + +- Provides two data storage layouts: + - Dense (flat vector) + - Sparse (hashmap based) + +- The components can be any type + - Crucially Symbolica `Atom` +- Can create tensor "networks" for deferred contraction + + +## Initializing + +Each tensor needs at minimum ae index `Structure`. +The `Structure` encodes the indices of the tensor, including the representation and dimensionality + +```rust +let mink = Representation::Lorentz(Dimension(4)); +let mu = Slot::from((AbstractIndex(0),mink)); +let bis = Representation::Bispinor(Dimension(4)); +let i = Slot::from((AbstractIndex(1),bis)); +let j = Slot::from((AbstractIndex(2),bis)); + +let structure = NamedStructure::new(&[mu,i,j],"γ"); +``` +Represents: +$$ +\gamma^{\mu}_{ij} +$$ + + + +## Adding data + +Now we can add data to this structure. +```{.rust code-line-numbers="|2|4|5"} +let iunit = Complex::::i(); +let mut gamma = SparseTensor::empty(structure); + +gamma.set(&[0, 0, 0], 1.); +gamma.set(&[1, 1, 0], 1.); +gamma.set(&[2, 2, 0], -1.); +gamma.set(&[3, 3, 0], -1.); + +gamma.set(&[0, 3, 1], 1.); +gamma.set(&[1, 2, 1], 1.); +gamma.set(&[2, 1, 1], -1.); +gamma.set(&[3, 0, 1], -1.); + +gamma.set(&[0, 3, 2], -iunit); +gamma.set(&[1, 2, 2], iunit); +gamma.set(&[2, 1, 2], iunit); +gamma.set(&[3, 0, 2], -iunit); + +gamma.set(&[0, 2, 3], 1.); +gamma.set(&[1, 3, 3], -1.); +gamma.set(&[2, 0, 3], -1.); +gamma.set(&[3, 1, 3], 1.); +``` + + +:::{.notes} + you can do this from a flattened vector, directly creating a dense tensor. You can also easily convert between the two storage options. +::: + +## What if I don't yet know the data: + +The tensor can be symbolic, or parametric! + +```{.rust code-line-numbers="|4|5|7|"} +let nu = Slot::from((AbstractIndex(1), mink)); +let structure = NamedStructure::from_slots(vec![mu, nu], "T"); + +let symbolicT: SymbolicTensor = SymbolicTensor::from_named(&structure).unwrap(); +println!("{}", symbolicT); // T(lor(4,0),lor(4,1)) +let paramT: DenseTensor = symbolicT.shadow().unwrap(); +let t12: Atom = paramT.get(&[1, 2]).unwrap());// T(1,2) +``` + +## What can you do with tensors: + +- Iterate along specific dimensions: + ```rust + for (c, _, _) in gamma.fiber(&[true, false, true].into()).iter() { + //stuff + } + ``` +- Arithmetic, when structures match: + ```rust + let 2gamma = gamma+gamma + ``` +- Contractions, when indices match: + ```rust + let cont = gamma.contract(¶mT); + ``` + +## Specific example {auto-animate=true } + +consider this triangle diagram: + +![](diag.svg){fig-align="center" width="30%"} + +The numerator spinor structure looks like + + + +$$p_1^{\mu_1} p_2^{\mu_2}p_3^{\mu_3} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_{\mu_2} \gamma_\rho \gamma_{\mu_3})$$ + + +:::{.notes} +what do we use it for? +::: + +## Specific example {auto-animate=true } + + +$$p_1^{\mu_1} p_2^{\mu_2}p_3^{\mu_3} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_{\mu_2} \gamma_\rho \gamma_{\mu_3})$$ + +Usually we turn this into dot products by repeatedly applying the defining relation: + +$$\left\{ \gamma_\mu, \gamma_\nu \right\} = \gamma_\mu \gamma_\nu + \gamma_\nu \gamma_\mu = 2 \eta_{\mu \nu} I_4$$ + + +## Specific example {auto-animate=true } + + +$$2 p_1^{\mu_1} p_2^{\mu_2}p_{3,\rho} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_{\mu_2}) \\ +-p_1^{\mu_1} p_2^{\mu_2}p_3^{\mu_3} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_{\mu_2} \gamma_{\mu_3}\gamma_\rho)$$ + +Usually we turn this into dot products by repeatedly applying the defining relation: + +$$\left\{ \gamma_\mu, \gamma_\nu \right\} = \gamma_\mu \gamma_\nu + \gamma_\nu \gamma_\mu = 2 \eta_{\mu \nu} I_4$$ + +## Specific example {auto-animate=true } + + +$$2 p_1^{\mu_1} p_2^{\mu_2}p_{3,\rho} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_{\mu_2}) \\ +-2 p_2 \cdot p_3 p_1^{\mu_1} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_\rho) \\ ++p_1^{\mu_1} p_2^{\mu_2}p_3^{\mu_3} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_{\mu_3}\gamma_{\mu_2} \gamma_\rho) +$$ + + +Usually we turn this into dot products by repeatedly applying the defining relation: + +$$\left\{ \gamma_\mu, \gamma_\nu \right\} = \gamma_\mu \gamma_\nu + \gamma_\nu \gamma_\mu = 2 \eta_{\mu \nu} I_4$$ + + +:::{.notes} +gamma mu 3 is parading down the length of the gamma chain,leaving a trail of shorter chains, and dot products +::: + +## Specific example {auto-animate=true } + + +$$-4(p_1 \cdot p_2) p_{3,\mu}\eta_{\nu\rho} + \\ +4(p_1\cdot p_2)p_{3,\nu}\eta_{\mu\rho} + \\ -4(p_1\cdot p_2)p_{3,\rho}\eta_{\mu\nu} + \\ +4(p_1\cdot p_3)p_{2,\mu}\eta_{\nu\rho} + \\ -4(p_1\cdot p_3)p_{2,\nu}\eta_{\mu\rho} + \\ -4(p_1\cdot p_3)p_{2,\rho}\eta_{\mu\nu} + \\ -4p_{1,\mu} (p_2\cdot p_3)\eta_{\nu\rho} + \\ +4p_{1,\mu} p_{2,\nu}p_{3,\rho} + \\ +4p_{1,\mu} p_{2,\rho}p_{3,\nu} + \\ -4p_{1,\nu}(p_2\cdot p_3)\eta_{\mu\rho} + \\ +4p_{1,\nu}p_{2,\mu}p_{3,\rho} + \\ +4p_{1,\nu}p_{2,\rho}p_{3,\mu} + \\ +4p_{1,\rho}(p_2\cdot p_3)\eta_{\mu\nu} + \\ -4p_{1,\rho}p_{2,\mu}p_{3,\nu} + \\ +4p_{1,\rho}p_{2,\nu}p_{3,\mu} +$$ + +## Problem + +Exponential growth of the terms: + + +::: {.nonincremental} +- length 6: 15 +- length 10: ~1000 +::: + +If you do componentwise contraction on the unmodified trace: + +$$p_1^{\mu_1} p_2^{\mu_2}p_3^{\mu_3} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_{\mu_2} \gamma_\rho \gamma_{\mu_3})$$ + +The scaling is linear in the chain length! + + +## Deffered contraction {auto-animate=true } + +Consider a string of tensors + +Example + +::: {data-id="gammachain"} + +$$\bar{v}_{i_1} u_{i_3}p_1^{\mu_1} p_2^{\mu_2}p_3^{\mu_3}p_4^{\mu_4}p_5^{\mu_5}p_6^{\mu_6}p_7^{\mu_7} \gamma_{i_1 i_2}^{\mu_1} +\\ \gamma_{i_2 i_3}^{\nu} +\gamma_{i_4 i_5}^{\nu} +\gamma_{i_5 i_6}^{\mu_2} +\gamma_{i_6 i_7}^{\mu_3} +\gamma_{i_7 i_8}^{\mu_4} +\gamma_{i_8 i_9}^{\mu_5} +\gamma_{i_9 i_4}^{\mu_6}$$ + +::: + + + + + +## Deffered contraction {auto-animate=true } + +We can model a string of tensors, with repeated indices as a graph, each edge is an index, each node a tensor. + +Example + +::: {data-id="gammachain"} + +```{dot .fragment} +//| echo: false +graph { node [shape=circle,height=0.1,label=""]; overlap="scale"; +layout=neato + 4294967299 -- 4294967297 [color=red] + 4294967299 -- 4294967298 + 4294967301 -- 4294967299 + 4294967301 -- 4294967300 + 4294967302 -- 4294967301 [color=red] + 4294967304 -- 4294967302 + 4294967304 -- 4294967303[color=red] + 4294967306 -- 4294967304 + 4294967306 -- 4294967305 [color=red] + 4294967308 -- 4294967306 + 4294967308 -- 4294967307 [color=red] + 4294967310 -- 4294967308 + 4294967310 -- 4294967309 [color=red] + 4294967311 -- 4294967310 + 4294967311 -- 4294967302 + 4294967312 -- 4294967311 [color=red]} +graph { node [shape=circle,height=0.1,label=""]; overlap="scale";} +``` +::: + +## Deffered contraction {auto-animate=true } + +We can model a string of tensors, with repeated indices as a graph, each edge is an index, each node a tensor. + +Example + +::: {data-id="gammachain"} + +```{.rust} +let γ1: MixedTensor<_> = gamma(1.into(), (1.into(), 2.into())).into(); +println!("{}", γ1.to_symbolic().unwrap()); +//γ(lor(4,1),spin(4,1),spina(4,2)) +let γ2: MixedTensor<_> = gamma(10.into(), (2.into(), 3.into())).into(); +// ... +let p1: MixedTensor<_> = param_mink_four_vector(1.into(), "p1").into(); +/// ... +let u: MixedTensor<_> = param_four_vector(1.into(), "u").into(); +println!("{}", u.to_symbolic().unwrap()); +// u(spin(4,1)) + +let net: TensorNetwork = TensorNetwork::from(vec![u,v,p1,p2, + p3,p4,p5,p6, + γ1,γ2,γ3,γ4, + γ5,γ6,γ7,γ8]); + +println!("{}",net.dot()); +``` +::: + +## Tensor Network + +::: {.columns} +::: {.column width="30%" } +```{dot .fragment} +//| echo: false +//| fig-width: 3 +graph { node [shape=circle,height=0.1,label=""]; overlap="scale"; +layout=neato + 4294967299 -- 4294967297 [color=red] + 4294967299 -- 4294967298 + 4294967301 -- 4294967299 + 4294967301 -- 4294967300 + 4294967302 -- 4294967301 [color=red] + 4294967304 -- 4294967302 + 4294967304 -- 4294967303[color=red] + 4294967306 -- 4294967304 + 4294967306 -- 4294967305 [color=red] + 4294967308 -- 4294967306 + 4294967308 -- 4294967307 [color=red] + 4294967310 -- 4294967308 + 4294967310 -- 4294967309 [color=red] + 4294967311 -- 4294967310 + 4294967311 -- 4294967302 + 4294967312 -- 4294967311 [color=red]} +graph { node [shape=circle,height=0.1,label=""]; overlap="scale";} +``` + +::: + +::: {.column width="70%"} + +- We now essentially have a tensor network + +- Contraction needs to happen according to some algorithm + + ```{.rust .smaller} + net.contract_algo(Self::edge_to_min_degree_node) + + ``` + +- This can be custom. + - Don't need to contract all the way down + + +::: +::: + + +## Shadowing {auto-animate=true auto-animate-id="shadowing" } + +::: {.nonincremental} +- If some tensors are fully parametric, there is a quick explosion of terms. +- We can contract until a certain limit, then "shadow" the obtained network with parametric tensors +::: + +:::::: {.columns} +::: {.column .fragment width="40%"} + +```{dot .fragment data-id="shadow"} +//| echo: false +//| fig-width: 3 +graph { node [shape=circle,height=0.1,label=""]; overlap="scale"; +layout=neato +edge [penwidth=10.0] +start=rsts + 4294967299 -- 4294967297 + 4294967299 -- 4294967298 + 4294967301 -- 4294967299 + 4294967301 -- 4294967300 + 4294967303 -- 4294967301 + 4294967303 -- 4294967302 + 4294967305 -- 4294967303 + 4294967305 -- 4294967304 + 4294967307 -- 4294967305 + 4294967307 -- 4294967306 + 4294967309 -- 4294967307 + 4294967309 -- 4294967308 + 4294967311 -- 4294967309 + 4294967311 -- 4294967310 + 4294967313 -- 4294967311 + 4294967313 -- 4294967312 + 4294967315 -- 4294967313 + 4294967315 -- 4294967314 + 4294967317 -- 4294967315 + 4294967317 -- 4294967316 + 4294967319 -- 4294967317 + 4294967319 -- 4294967318 + 4294967321 -- 4294967319 + 4294967321 -- 4294967320 + 4294967323 -- 4294967321 + 4294967323 -- 4294967322 + 4294967325 -- 4294967323 + 4294967325 -- 4294967324 + 4294967327 -- 4294967325 + 4294967327 -- 4294967326 + 4294967329 -- 4294967327 + 4294967329 -- 4294967328 + 4294967331 -- 4294967329 + 4294967331 -- 4294967330 + 4294967333 -- 4294967331 + 4294967333 -- 4294967332 + 4294967335 -- 4294967333 + 4294967335 -- 4294967334 + 4294967337 -- 4294967335 + 4294967337 -- 4294967336 + 4294967338 -- 4294967337 + 4294967340 -- 4294967338 + 4294967340 -- 4294967339 + 4294967342 -- 4294967340 + 4294967342 -- 4294967341 + 4294967344 -- 4294967342 + 4294967344 -- 4294967343 + 4294967346 -- 4294967344 + 4294967346 -- 4294967345 + 4294967348 -- 4294967346 + 4294967348 -- 4294967347 + 4294967350 -- 4294967348 + 4294967350 -- 4294967349 + 4294967352 -- 4294967350 + 4294967352 -- 4294967351 + 4294967354 -- 4294967352 + 4294967354 -- 4294967353 + 4294967356 -- 4294967354 + 4294967356 -- 4294967355 + 4294967358 -- 4294967356 + 4294967358 -- 4294967357 + 4294967360 -- 4294967358 + 4294967360 -- 4294967359 + 4294967362 -- 4294967360 + 4294967362 -- 4294967361 + 4294967364 -- 4294967362 + 4294967364 -- 4294967363 + 4294967366 -- 4294967364 + 4294967366 -- 4294967365 + 4294967368 -- 4294967366 + 4294967368 -- 4294967367 + 4294967370 -- 4294967368 + 4294967370 -- 4294967369 + 4294967372 -- 4294967370 + 4294967372 -- 4294967371 + 4294967374 -- 4294967372 + 4294967374 -- 4294967373 + 4294967376 -- 4294967374 + 4294967376 -- 4294967375 + 4294967378 -- 4294967376 + 4294967378 -- 4294967377 + 4294967380 -- 4294967378 + 4294967380 -- 4294967379 + 4294967382 -- 4294967380 + 4294967382 -- 4294967381 + 4294967384 -- 4294967382 + 4294967384 -- 4294967383 + 4294967386 -- 4294967384 + 4294967386 -- 4294967385 + 4294967387 -- 4294967386 + 4294967387 -- 4294967338 + 4294967388 -- 4294967387} +``` + +::: +::: {.column .fragment width="30%"} +```{dot .fragment data-id="shadow"} +//| echo: false +//| fig-width: 3 +graph { node [shape=circle,height=0.1,label=""]; overlap="scale"; +edge [penwidth=3.0] +start=trst +layout=neato + 30064771165 -- 12884901893 + 21474836491 -- 30064771165 + 21474836497 -- 21474836491 + 21474836503 -- 21474836497 + 21474836509 -- 21474836503 + 21474836515 -- 21474836509 + 12884901930 -- 21474836515 + 21474836526 -- 12884901930 + 21474836532 -- 21474836526 + 21474836538 -- 21474836532 + 21474836544 -- 21474836538 + 21474836550 -- 21474836544 + 21474836556 -- 21474836550 + 21474836562 -- 21474836556 + 12884901976 -- 21474836562 + 12884901978 -- 12884901976 + 12884901978 -- 12884901930} +``` +::: +::: {.column .fragment width="30%"} + +```{dot .fragment data-id="shadow"} +//| echo: false +//| fig-width: 3 + +graph { node [shape=circle,height=0.1,label=""]; overlap="scale"; +edge [penwidth=2.0] +start=st +layout=neato + 38654705757 -- 133143986267 + 30064771089 -- 38654705757 + 30064771101 -- 30064771089 + 21474836522 -- 30064771101 + 30064771124 -- 21474836522 + 30064771136 -- 30064771124 + 30064771148 -- 30064771136 + 12884901978 -- 30064771148 + 12884901978 -- 30064771101} +``` +::: +:::::: + +- A similar idea has already been done in cuTensorNet! + +## Synergy with symbolica + +Spenso understands a specific format of symbolica expressions and can recognise known tensors with numerical counterparts: + +```rust +let structure = NamedStructure::from_slots(vec![mu, i, j], "γ"); +let p_struct = NamedStructure::from_slots(vec![mu], "p"); +let t_struct = NamedStructure::from_slots(vec![i, j, k], "T"); + +let gamma_sym = SymbolicTensor::from_named(&structure).unwrap(); +let p_sym = SymbolicTensor::from_named(&p_struct).unwrap(); +let t_sym = SymbolicTensor::from_named(&t_struct).unwrap(); + +let f = gamma_sym.contract(&p_sym).unwrap().contract(&t_sym).unwrap(); + +println!("{}", *f.get_atom()); +``` +```python +p(lor(4,0))*γ(lor(4,0),bis(4,1),bis(4,2))*T(bis(4,1),bis(4,2),bis(4,3)) +``` + +```rust +let net: TensorNetwork = f.to_network().unwrap(); +``` + +```{dot} +graph { node [shape=circle,height=0.1,label=""]; overlap="scale"; +layout=neato + 4294967298 -- 4294967297 + 4294967299 -- 4294967298 + 4294967299 -- 4294967298ext4294967303 [shape=none, label=""]; + 4294967299 -- ext4294967303;} +``` + +## Performance + + +:::::: {.columns} +::: {.column } +```{dot .fragment data-id="shadow"} +//| echo: false +//| fig-width: 2 +graph { node [shape=circle,height=0.1,label=""]; overlap="scale"; +edge [penwidth=10.0] + +layout=neato + 4294967299 -- 4294967297 + 4294967299 -- 4294967298 + 4294967301 -- 4294967299 + 4294967301 -- 4294967300 + 4294967303 -- 4294967301 + 4294967303 -- 4294967302 + 4294967305 -- 4294967303 + 4294967305 -- 4294967304 + 4294967307 -- 4294967305 + 4294967307 -- 4294967306 + 4294967309 -- 4294967307 + 4294967309 -- 4294967308 + 4294967311 -- 4294967309 + 4294967311 -- 4294967310 + 4294967313 -- 4294967311 + 4294967313 -- 4294967312 + 4294967315 -- 4294967313 + 4294967315 -- 4294967314 + 4294967317 -- 4294967315 + 4294967317 -- 4294967316 + 4294967319 -- 4294967317 + 4294967319 -- 4294967318 + 4294967321 -- 4294967319 + 4294967321 -- 4294967320 + 4294967323 -- 4294967321 + 4294967323 -- 4294967322 + 4294967325 -- 4294967323 + 4294967325 -- 4294967324 + 4294967327 -- 4294967325 + 4294967327 -- 4294967326 + 4294967329 -- 4294967327 + 4294967329 -- 4294967328 + 4294967331 -- 4294967329 + 4294967331 -- 4294967330 + 4294967333 -- 4294967331 + 4294967333 -- 4294967332 + 4294967335 -- 4294967333 + 4294967335 -- 4294967334 + 4294967337 -- 4294967335 + 4294967337 -- 4294967336 + 4294967338 -- 4294967337 + 4294967340 -- 4294967338 + 4294967340 -- 4294967339 + 4294967342 -- 4294967340 + 4294967342 -- 4294967341 + 4294967344 -- 4294967342 + 4294967344 -- 4294967343 + 4294967346 -- 4294967344 + 4294967346 -- 4294967345 + 4294967348 -- 4294967346 + 4294967348 -- 4294967347 + 4294967350 -- 4294967348 + 4294967350 -- 4294967349 + 4294967352 -- 4294967350 + 4294967352 -- 4294967351 + 4294967354 -- 4294967352 + 4294967354 -- 4294967353 + 4294967356 -- 4294967354 + 4294967356 -- 4294967355 + 4294967358 -- 4294967356 + 4294967358 -- 4294967357 + 4294967360 -- 4294967358 + 4294967360 -- 4294967359 + 4294967362 -- 4294967360 + 4294967362 -- 4294967361 + 4294967364 -- 4294967362 + 4294967364 -- 4294967363 + 4294967366 -- 4294967364 + 4294967366 -- 4294967365 + 4294967368 -- 4294967366 + 4294967368 -- 4294967367 + 4294967370 -- 4294967368 + 4294967370 -- 4294967369 + 4294967372 -- 4294967370 + 4294967372 -- 4294967371 + 4294967374 -- 4294967372 + 4294967374 -- 4294967373 + 4294967376 -- 4294967374 + 4294967376 -- 4294967375 + 4294967378 -- 4294967376 + 4294967378 -- 4294967377 + 4294967380 -- 4294967378 + 4294967380 -- 4294967379 + 4294967382 -- 4294967380 + 4294967382 -- 4294967381 + 4294967384 -- 4294967382 + 4294967384 -- 4294967383 + 4294967386 -- 4294967384 + 4294967386 -- 4294967385 + 4294967387 -- 4294967386 + 4294967387 -- 4294967338 + 4294967388 -- 4294967387} + ``` +::: +::: {.column} + +| Spenso | Spenso Compiled | Hardcoded Fortran | +|---------|:-----:|:------:| +| 104 μs | 16 μs | 31μs | + +We can use a compiled version of the stacked shadowing using symbolica! + +::: +:::::: +# Outlook + +- More tooling to deal with specific tensor index symmetries +- Python bindings compatible with symbolica +- GPU support +- SIMD support + + +# Thanks + +If you want to check it out it is on crates.io: + + + +And on github: + + \ No newline at end of file diff --git a/pres/spenso.png b/pres/spenso.png new file mode 100644 index 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z55Hhxi(+$SW%mD!T6(>eHU0Ld$scG+0xMZ)tb0N`#QNo_onZi|%yLMB869;BU;#Ox zP^g=^N*B;m^+=_Jf_9TrYST0Qdr-`{JQ}y-0sjx|W`CXGAV`^NSI~C!N9xT=+z-$HJc>4EQK&O@(p_^l@yuVZXNU zE$#qw(@I2L_-?QGFa9p9yQWD|De#^ zUYR$z9kKI#3G`E>;7ZDAJAHk9xu>Y2Ozy?S#a2-_&O|13+F&|ETj0fDK-l(|$7Ue~ z?bs^|PsHV0(!1qPi9|^+R)l`>6NIHgVM~-*>nW+RR;pJ>v7q<{dSU)+f=Wha=21l{ zwQ&Nl)k^ZIpV{g)mx=p4sdUco&-n1`tV~#d>wec?;eV#Egyra;TI&QJHlSzapt;wj|dyDl3q1{NkTOK#9fN zP+j1Foi#pAC+&D#Q^C7@`%vf;Qz z=B*UndTM(5)HmgzGXNfODUQFUrbZTyBEn=x_;z%v2@Xa_y`7|fSM@`OoiQ>HpxOU$ zQmea5Ca?##GHnKrw+P8tx)gMyn05#M@$?+;S->sncML5JvSpUlDXeosT7uLqe&B#K zDJe+-qY@IVVSMpBzNx9{4f3Pv^^Kv$lc?qpV39$~0WDHdhj8JVHW@gRzqVPs(jN>a eXIDM`SmgmcdQYvnCkk+hPV15G!*VsN(EkD::i(); + let unit = Complex::::from(1.); + let mut gamma = SparseTensor::empty(structure); + + gamma.set(&[0, 0, 0], unit).unwrap(); + gamma.set(&[1, 1, 0], unit).unwrap(); + gamma.set(&[2, 2, 0], -unit).unwrap(); + gamma.set(&[3, 3, 0], -unit).unwrap(); + + gamma.set(&[0, 3, 1], unit).unwrap(); + gamma.set(&[1, 2, 1], unit).unwrap(); + gamma.set(&[2, 1, 1], -unit).unwrap(); + gamma.set(&[3, 0, 1], -unit).unwrap(); + + gamma.set(&[0, 3, 2], -iunit).unwrap(); + gamma.set(&[1, 2, 2], iunit).unwrap(); + gamma.set(&[2, 1, 2], iunit).unwrap(); + gamma.set(&[3, 0, 2], -iunit).unwrap(); + + gamma.set(&[0, 2, 3], unit).unwrap(); + gamma.set(&[1, 3, 3], -unit).unwrap(); + gamma.set(&[2, 0, 3], -unit).unwrap(); + gamma.set(&[3, 1, 3], unit).unwrap(); + + let nu = Slot::from((AbstractIndex(1), mink)); + let structure = NamedStructure::from_slots(vec![mu, nu], "T"); + + let symbolicT: SymbolicTensor = SymbolicTensor::from_named(&structure).unwrap(); + println!("{}", symbolicT); // T(lor(4,0),lor(4,1)) + let paramT: DenseTensor = symbolicT.shadow().unwrap(); + + println!("{}", paramT.get(&[1, 2]).unwrap()); + + for (c, _, _) in gamma.fiber([true, false, true].as_slice().into()).iter() { + //stuff + } +}

+
+ +
+

Spenso

+

a rust-based framework for symbolic and numerical tensor computation

+ +
+
+
+Lucien Huber +
+

+ Bern University +

+
+
+ +
+
+
+

Introduction

+

Tensors are pervasive in High Energy Physics.

+

We wanted a library that implements tensor contractions, and arithmetic, component wise and synergises well with symbolica.

+ +
+
+

Uses

+

Evaluating:

+
    +
  • Feynman Rules (e.g. gamma chains)
    • +
  • IBP (multi-dim. array)
    • +
  • Tensor Networks for lattice QCD
    • +
+ +
+
+

It already exists, no?

+

Not really

+
    +
  • Mathematica +
      +
    • Too slow
      • +
    • +
  • NDarray (Rust) +
      +
    • Wanted Einstein/Abstract Index notation
      • +
    • Needed Generic Dimensions
      • +
    • +
  • cuTensorNet (CUDA) +
      +
    • Support for non-euclidean metric
      • +
    • Physical indices
      • +
    • +
  • ITensors.jl (Julia/C++) +
      +
    • Wanted Rust (for LU/gammaloop), zero-cost abstractions FTW
      • +
    • Does not talk Symbolica
      • +
    • +
+ +
+
+
+

Introducing Spenso!

+ +
    +
  • Provides two data storage layouts: +
      +
    • Dense (flat vector)
      • +
    • Sparse (hashmap based)
      • +
    • +
  • The components can be any type +
      +
    • Crucially Symbolica Atom
      • +
    • +
  • Can create tensor “networks” for deferred contraction
    • +
+
+
+

Initializing

+

Each tensor needs at minimum ae index Structure. The Structure encodes the indices of the tensor, including the representation and dimensionality

+
let mink = Representation::Lorentz(Dimension(4));
+let mu = Slot::from((AbstractIndex(0),mink));
+let bis = Representation::Bispinor(Dimension(4));
+let i = Slot::from((AbstractIndex(1),bis));
+let j = Slot::from((AbstractIndex(2),bis));
+
+let structure = NamedStructure::new(&[mu,i,j],"γ");
+

Represents: \[ +\gamma^{\mu}_{ij} +\]

+
+
+

Adding data

+

Now we can add data to this structure.

+
let iunit = Complex::<f64>::i();
+let mut gamma = SparseTensor::empty(structure);
+
+gamma.set(&[0, 0, 0], 1.);
+gamma.set(&[1, 1, 0], 1.);
+gamma.set(&[2, 2, 0], -1.);
+gamma.set(&[3, 3, 0], -1.);
+
+gamma.set(&[0, 3, 1], 1.);
+gamma.set(&[1, 2, 1], 1.);
+gamma.set(&[2, 1, 1], -1.);
+gamma.set(&[3, 0, 1], -1.);
+
+gamma.set(&[0, 3, 2], -iunit);
+gamma.set(&[1, 2, 2], iunit);
+gamma.set(&[2, 1, 2], iunit);
+gamma.set(&[3, 0, 2], -iunit);
+
+gamma.set(&[0, 2, 3], 1.);
+gamma.set(&[1, 3, 3], -1.);
+gamma.set(&[2, 0, 3], -1.);
+gamma.set(&[3, 1, 3], 1.);
+ +
+
+

What if I don’t yet know the data:

+

The tensor can be symbolic, or parametric!

+
let nu = Slot::from((AbstractIndex(1), mink));
+let structure = NamedStructure::from_slots(vec![mu, nu], "T");
+
+let symbolicT: SymbolicTensor = SymbolicTensor::from_named(&structure).unwrap();
+println!("{}", symbolicT); // T(lor(4,0),lor(4,1))
+let paramT: DenseTensor<Atom, _> = symbolicT.shadow().unwrap();
+let t12: Atom = paramT.get(&[1, 2]).unwrap());// T(1,2)
+
+
+

What can you do with tensors:

+
    +

  • Iterate along specific dimensions:

    +
    for (c, _, _) in gamma.fiber(&[true, false, true].into()).iter() {
+        //stuff
+}
    • +

  • Arithmetic, when structures match:

    +
      let 2gamma = gamma+gamma
    • +

  • Contractions, when indices match:

    +
      let cont = gamma.contract(&paramT);
    • +
+
+
+

Specific example

+

consider this triangle diagram:

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

The numerator spinor structure looks like

+

\[p_1^{\mu_1} p_2^{\mu_2}p_3^{\mu_3} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_{\mu_2} \gamma_\rho \gamma_{\mu_3})\]

+ +
+
+

Specific example

+

\[p_1^{\mu_1} p_2^{\mu_2}p_3^{\mu_3} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_{\mu_2} \gamma_\rho \gamma_{\mu_3})\]

+

Usually we turn this into dot products by repeatedly applying the defining relation:

+

\[\left\{ \gamma_\mu, \gamma_\nu \right\} = \gamma_\mu \gamma_\nu + \gamma_\nu \gamma_\mu = 2 \eta_{\mu \nu} I_4\]

+
+
+

Specific example

+

\[2 p_1^{\mu_1} p_2^{\mu_2}p_{3,\rho} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_{\mu_2}) \\ +-p_1^{\mu_1} p_2^{\mu_2}p_3^{\mu_3} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_{\mu_2} \gamma_{\mu_3}\gamma_\rho)\]

+

Usually we turn this into dot products by repeatedly applying the defining relation:

+

\[\left\{ \gamma_\mu, \gamma_\nu \right\} = \gamma_\mu \gamma_\nu + \gamma_\nu \gamma_\mu = 2 \eta_{\mu \nu} I_4\]

+
+
+

Specific example

+

\[2 p_1^{\mu_1} p_2^{\mu_2}p_{3,\rho} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_{\mu_2}) \\ +-2 p_2 \cdot p_3 p_1^{\mu_1} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_\rho) \\ ++p_1^{\mu_1} p_2^{\mu_2}p_3^{\mu_3} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_{\mu_3}\gamma_{\mu_2} \gamma_\rho) +\]

+

Usually we turn this into dot products by repeatedly applying the defining relation:

+

\[\left\{ \gamma_\mu, \gamma_\nu \right\} = \gamma_\mu \gamma_\nu + \gamma_\nu \gamma_\mu = 2 \eta_{\mu \nu} I_4\]

+ +
+
+

Specific example

+

\[-4(p_1 \cdot p_2) p_{3,\mu}\eta_{\nu\rho} + \\ +4(p_1\cdot p_2)p_{3,\nu}\eta_{\mu\rho} + \\ -4(p_1\cdot p_2)p_{3,\rho}\eta_{\mu\nu} + \\ +4(p_1\cdot p_3)p_{2,\mu}\eta_{\nu\rho} + \\ -4(p_1\cdot p_3)p_{2,\nu}\eta_{\mu\rho} + \\ -4(p_1\cdot p_3)p_{2,\rho}\eta_{\mu\nu} + \\ -4p_{1,\mu} (p_2\cdot p_3)\eta_{\nu\rho} + \\ +4p_{1,\mu} p_{2,\nu}p_{3,\rho} + \\ +4p_{1,\mu} p_{2,\rho}p_{3,\nu} + \\ -4p_{1,\nu}(p_2\cdot p_3)\eta_{\mu\rho} + \\ +4p_{1,\nu}p_{2,\mu}p_{3,\rho} + \\ +4p_{1,\nu}p_{2,\rho}p_{3,\mu} + \\ +4p_{1,\rho}(p_2\cdot p_3)\eta_{\mu\nu} + \\ -4p_{1,\rho}p_{2,\mu}p_{3,\nu} + \\ +4p_{1,\rho}p_{2,\nu}p_{3,\mu} +\]

+
+
+

Problem

+

Exponential growth of the terms:

+
+
    +
  • length 6: 15
    • +
  • length 10: ~1000
    • +
+
+

If you do componentwise contraction on the unmodified trace:

+

\[p_1^{\mu_1} p_2^{\mu_2}p_3^{\mu_3} \mathrm{Tr} (\gamma_\mu \gamma_{\mu_1} \gamma_\nu \gamma_{\mu_2} \gamma_\rho \gamma_{\mu_3})\]

+

The scaling is linear in the chain length!

+
+
+

Deffered contraction

+

Consider a string of tensors

+

Example

+
+

\[\bar{v}_{i_1} u_{i_3}p_1^{\mu_1} p_2^{\mu_2}p_3^{\mu_3}p_4^{\mu_4}p_5^{\mu_5}p_6^{\mu_6}p_7^{\mu_7} \gamma_{i_1 i_2}^{\mu_1} +\\ \gamma_{i_2 i_3}^{\nu} +\gamma_{i_4 i_5}^{\nu} +\gamma_{i_5 i_6}^{\mu_2} +\gamma_{i_6 i_7}^{\mu_3} +\gamma_{i_7 i_8}^{\mu_4} +\gamma_{i_8 i_9}^{\mu_5} +\gamma_{i_9 i_4}^{\mu_6}\]

+
+
+
+

Deffered contraction

+

We can model a string of tensors, with repeated indices as a graph, each edge is an index, each node a tensor.

+

Example

+
+
+
+
+

+
+ + + + + +4294967299 + + + + +4294967297 + + + + +4294967299--4294967297 + + + + +4294967298 + + + + +4294967299--4294967298 + + + + +4294967301 + + + + +4294967301--4294967299 + + + + +4294967300 + + + + +4294967301--4294967300 + + + + +4294967302 + + + + +4294967302--4294967301 + + + + +4294967304 + + + + +4294967304--4294967302 + + + + +4294967303 + + + + +4294967304--4294967303 + + + + +4294967306 + + + + +4294967306--4294967304 + + + + +4294967305 + + + + +4294967306--4294967305 + + + + +4294967308 + + + + +4294967308--4294967306 + + + + +4294967307 + + + + +4294967308--4294967307 + + + + +4294967310 + + + + +4294967310--4294967308 + + + + +4294967309 + + + + +4294967310--4294967309 + + + + +4294967311 + + + + +4294967311--4294967302 + + + + +4294967311--4294967310 + + + + +4294967312 + + + + +4294967312--4294967311 + + + + +
+

+
+
+
+
+
+
+

Deffered contraction

+

We can model a string of tensors, with repeated indices as a graph, each edge is an index, each node a tensor.

+

Example

+
+
let γ1: MixedTensor<_> = gamma(1.into(), (1.into(), 2.into())).into();
+println!("{}", γ1.to_symbolic().unwrap()); 
+//γ(lor(4,1),spin(4,1),spina(4,2))
+let γ2: MixedTensor<_> = gamma(10.into(), (2.into(), 3.into())).into();
+// ...
+let p1: MixedTensor<_> = param_mink_four_vector(1.into(), "p1").into();
+/// ...
+let u: MixedTensor<_> = param_four_vector(1.into(), "u").into();
+println!("{}", u.to_symbolic().unwrap()); 
+// u(spin(4,1))
+
+let net: TensorNetwork<NumTensor> = TensorNetwork::from(vec![u,v,p1,p2,
+                                                            p3,p4,p5,p6,
+                                                            γ1,γ2,γ3,γ4,
+                                                            γ5,γ6,γ7,γ8]);
+
+println!("{}",net.dot());
+
+
+
+

Tensor Network

+
+
+
+
+
+

+
+ + + + + +4294967299 + + + + +4294967297 + + + + +4294967299--4294967297 + + + + +4294967298 + + + + +4294967299--4294967298 + + + + +4294967301 + + + + +4294967301--4294967299 + + + + +4294967300 + + + + +4294967301--4294967300 + + + + +4294967302 + + + + +4294967302--4294967301 + + + + +4294967304 + + + + +4294967304--4294967302 + + + + +4294967303 + + + + +4294967304--4294967303 + + + + +4294967306 + + + + +4294967306--4294967304 + + + + +4294967305 + + + + +4294967306--4294967305 + + + + +4294967308 + + + + +4294967308--4294967306 + + + + +4294967307 + + + + +4294967308--4294967307 + + + + +4294967310 + + + + +4294967310--4294967308 + + + + +4294967309 + + + + +4294967310--4294967309 + + + + +4294967311 + + + + +4294967311--4294967302 + + + + +4294967311--4294967310 + + + + +4294967312 + + + + +4294967312--4294967311 + + + + +
+

+
+
+
+
+
    +

  • We now essentially have a tensor network

    • +

  • Contraction needs to happen according to some algorithm

    +
    net.contract_algo(Self::edge_to_min_degree_node)
    • +

  • This can be custom.

    • +

  • Don’t need to contract all the way down

    • +
+
+
+
+
+

Shadowing

+
+
    +
  • If some tensors are fully parametric, there is a quick explosion of terms.
    • +
  • We can contract until a certain limit, then “shadow” the obtained network with parametric tensors
    • +
+
+
+
+
+
+
+

+
+ + + + + +4294967299 + + + + +4294967297 + + + + +4294967299--4294967297 + + + + +4294967298 + + + + +4294967299--4294967298 + + + + +4294967301 + + + + +4294967301--4294967299 + + + + +4294967300 + + + + +4294967301--4294967300 + + + + +4294967303 + + + + +4294967303--4294967301 + + + + +4294967302 + + + + +4294967303--4294967302 + + + + +4294967305 + + + + +4294967305--4294967303 + + + + +4294967304 + + + + +4294967305--4294967304 + + + + +4294967307 + + + + +4294967307--4294967305 + + + + +4294967306 + + + + +4294967307--4294967306 + + + + +4294967309 + + + + +4294967309--4294967307 + + + + +4294967308 + + + + +4294967309--4294967308 + + + + +4294967311 + + + + +4294967311--4294967309 + + + + +4294967310 + + + + +4294967311--4294967310 + + + + +4294967313 + + + + +4294967313--4294967311 + + + + +4294967312 + + + + +4294967313--4294967312 + + + + +4294967315 + + + + +4294967315--4294967313 + + + + +4294967314 + + + + +4294967315--4294967314 + + + + +4294967317 + + + + +4294967317--4294967315 + + + + +4294967316 + + + + +4294967317--4294967316 + + + + +4294967319 + + + + +4294967319--4294967317 + + + + +4294967318 + + + + +4294967319--4294967318 + + + + +4294967321 + + + + +4294967321--4294967319 + + + + +4294967320 + + + + +4294967321--4294967320 + + + + +4294967323 + + + + +4294967323--4294967321 + + + + +4294967322 + + + + +4294967323--4294967322 + + + + +4294967325 + + + + +4294967325--4294967323 + + + + +4294967324 + + + + +4294967325--4294967324 + + + + +4294967327 + + + + +4294967327--4294967325 + + + + +4294967326 + + + + +4294967327--4294967326 + + + + +4294967329 + + + + +4294967329--4294967327 + + + + +4294967328 + + + + +4294967329--4294967328 + + + + +4294967331 + + + + +4294967331--4294967329 + + + + +4294967330 + + + + +4294967331--4294967330 + + + + +4294967333 + + + + +4294967333--4294967331 + + + + +4294967332 + + + + +4294967333--4294967332 + + + + +4294967335 + + + + +4294967335--4294967333 + + + + +4294967334 + + + + +4294967335--4294967334 + + + + +4294967337 + + + + +4294967337--4294967335 + + + + +4294967336 + + + + +4294967337--4294967336 + + + + +4294967338 + + + + +4294967338--4294967337 + + + + +4294967340 + + + + +4294967340--4294967338 + + + + +4294967339 + + + + +4294967340--4294967339 + + + + +4294967342 + + + + +4294967342--4294967340 + + + + +4294967341 + + + + +4294967342--4294967341 + + + + +4294967344 + + + + +4294967344--4294967342 + + + + +4294967343 + + + + +4294967344--4294967343 + + + + +4294967346 + + + + +4294967346--4294967344 + + + + +4294967345 + + + + +4294967346--4294967345 + + + + +4294967348 + + + + +4294967348--4294967346 + + + + +4294967347 + + + + +4294967348--4294967347 + + + + +4294967350 + + + + +4294967350--4294967348 + + + + +4294967349 + + + + +4294967350--4294967349 + + + + +4294967352 + + + + +4294967352--4294967350 + + + + +4294967351 + + + + +4294967352--4294967351 + + + + +4294967354 + + + + +4294967354--4294967352 + + + + +4294967353 + + + + +4294967354--4294967353 + + + + +4294967356 + + + + +4294967356--4294967354 + + + + +4294967355 + + + + +4294967356--4294967355 + + + + +4294967358 + + + + +4294967358--4294967356 + + + + +4294967357 + + + + +4294967358--4294967357 + + + + +4294967360 + + + + +4294967360--4294967358 + + + + +4294967359 + + + + +4294967360--4294967359 + + + + +4294967362 + + + + +4294967362--4294967360 + + + + +4294967361 + + + + +4294967362--4294967361 + + + + +4294967364 + + + + +4294967364--4294967362 + + + + +4294967363 + + + + +4294967364--4294967363 + + + + +4294967366 + + + + +4294967366--4294967364 + + + + +4294967365 + + + + +4294967366--4294967365 + + + + +4294967368 + + + + +4294967368--4294967366 + + + + +4294967367 + + + + +4294967368--4294967367 + + + + +4294967370 + + + + +4294967370--4294967368 + + + + +4294967369 + + + + +4294967370--4294967369 + + + + +4294967372 + + + + +4294967372--4294967370 + + + + +4294967371 + + + + +4294967372--4294967371 + + + + +4294967374 + + + + +4294967374--4294967372 + + + + +4294967373 + + + + +4294967374--4294967373 + + + + +4294967376 + + + + +4294967376--4294967374 + + + + +4294967375 + + + + +4294967376--4294967375 + + + + +4294967378 + + + + +4294967378--4294967376 + + + + +4294967377 + + + + +4294967378--4294967377 + + + + +4294967380 + + + + +4294967380--4294967378 + + + + +4294967379 + + + + +4294967380--4294967379 + + + + +4294967382 + + + + +4294967382--4294967380 + + + + +4294967381 + + + + +4294967382--4294967381 + + + + +4294967384 + + + + +4294967384--4294967382 + + + + +4294967383 + + + + +4294967384--4294967383 + + + + +4294967386 + + + + +4294967386--4294967384 + + + + +4294967385 + + + + +4294967386--4294967385 + + + + +4294967387 + + + + +4294967387--4294967338 + + + + +4294967387--4294967386 + + + + +4294967388 + + + + +4294967388--4294967387 + + + + +
+

+
+
+
+
+
+
+
+

+
+ + + + + +30064771165 + + + + +12884901893 + + + + +30064771165--12884901893 + + + + +21474836491 + + + + +21474836491--30064771165 + + + + +21474836497 + + + + +21474836497--21474836491 + + + + +21474836503 + + + + +21474836503--21474836497 + + + + +21474836509 + + + + +21474836509--21474836503 + + + + +21474836515 + + + + +21474836515--21474836509 + + + + +12884901930 + + + + +12884901930--21474836515 + + + + +21474836526 + + + + +21474836526--12884901930 + + + + +21474836532 + + + + +21474836532--21474836526 + + + + +21474836538 + + + + +21474836538--21474836532 + + + + +21474836544 + + + + +21474836544--21474836538 + + + + +21474836550 + + + + +21474836550--21474836544 + + + + +21474836556 + + + + +21474836556--21474836550 + + + + +21474836562 + + + + +21474836562--21474836556 + + + + +12884901976 + + + + +12884901976--21474836562 + + + + +12884901978 + + + + +12884901978--12884901930 + + + + +12884901978--12884901976 + + + + +
+

+
+
+
+
+
+
+
+

+
+ + + + + +38654705757 + + + + +133143986267 + + + + +38654705757--133143986267 + + + + +30064771089 + + + + +30064771089--38654705757 + + + + +30064771101 + + + + +30064771101--30064771089 + + + + +21474836522 + + + + +21474836522--30064771101 + + + + +30064771124 + + + + +30064771124--21474836522 + + + + +30064771136 + + + + +30064771136--30064771124 + + + + +30064771148 + + + + +30064771148--30064771136 + + + + +12884901978 + + + + +12884901978--30064771101 + + + + +12884901978--30064771148 + + + + +
+

+
+
+
+
+
+
    +
  • A similar idea has already been done in cuTensorNet!
    • +
+
+
+

Synergy with symbolica

+

Spenso understands a specific format of symbolica expressions and can recognise known tensors with numerical counterparts:

+
let structure = NamedStructure::from_slots(vec![mu, i, j], "γ");
+let p_struct = NamedStructure::from_slots(vec![mu], "p");
+let t_struct = NamedStructure::from_slots(vec![i, j, k], "T");
+
+let gamma_sym = SymbolicTensor::from_named(&structure).unwrap();
+let p_sym = SymbolicTensor::from_named(&p_struct).unwrap();
+let t_sym = SymbolicTensor::from_named(&t_struct).unwrap();
+
+let f = gamma_sym.contract(&p_sym).unwrap().contract(&t_sym).unwrap();
+
+println!("{}", *f.get_atom());
+
p(lor(4,0))*γ(lor(4,0),bis(4,1),bis(4,2))*T(bis(4,1),bis(4,2),bis(4,3))
+
let net: TensorNetwork<MixedTensor> = f.to_network().unwrap();
+
+
+
+

+
+ + + + + +4294967298 + + + + +4294967297 + + + + +4294967298--4294967297 + + + + +4294967299 + + + + +4294967299--4294967298 + + + + +4294967299--4294967298 + + + + +ext4294967303 + + + +4294967299--ext4294967303 + + + + +
+

+
+
+
+
+
+

Performance

+
+
+
+
+
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SpensoSpenso CompiledHardcoded Fortran
104 μs16 μs31μs
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We can use a compiled version of the stacked shadowing using symbolica!

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Outlook

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  • More tooling to deal with specific tensor index symmetries
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  • Python bindings compatible with symbolica
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  • GPU support
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  • SIMD support
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Thanks

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If you want to check it out it is on crates.io:

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https://crates.io/crates/spenso

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And on github:

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https://github.com/alphal00p/spenso

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