Improve text of the paper

paper.md

@@ -11,14 +11,11 @@ authors:
   - name: Luisa Orozco
     orcid: 0000-0002-9153-650X
     corresponding: true # corresponding author
-    equal-contrib: true
     affiliation: 1
   - name: Aron Jansen
     orcid: 0000-0002-4764-9347
-    equal-contrib: true
     affiliation: 1
   - name: Retief Lubbe
-    equal-contrib: true
     affiliation: 2
   - name: Hongyang Cheng
     orcid: 0000-0001-7652-8600
@@ -37,25 +34,33 @@ bibliography: paper.bib
 
 How to keep dikes safe with rising sea levels? Why are ripples formed in sand? What can we prepare for landing on Mars? At the center of these questions is the understanding of how the grains, as a self-organizing material, collide, flow, or get jammed and compressed. State-of-the-art algorithms allow for simulating millions of grains individually in a computer. However, such computations can take very long and produce complex data difficult to interpret and be upscaled to large-scale applications such as sediment transport and debris flows. GrainLearning is an open-source toolbox with machine learning and statistical inference modules allowing for emulating granular material behavior and learning material uncertainties from real-life observations.
 
+To understand what GrainLearning does, let us consider a mechanical test performed on a granular material. The macroscopic response of such material, in terms of stress-strain evolution curves, is obtained from the test.
+It would be interesting to have a digital equivalent material to further investigate, using numerical simulations such as the discrete element method (DEM), how such material would behave under other mechanical constraints. To do so, the first step is defining a contact model governing interactions between grains in DEM. Then, using GrainLearning one can calibrate or infer the values of the contact model parameters such that the mechanical response observed in the real-world experiment is the closest to the one obtained in the DEM simulation.
+
+Making abstraction of this idea, GrainLearning's use can be extended to other kind of models or *Dynamical systems* that can be other simulation frameworks such as FEM, CFD, LBM, and even other techniques such as agent based modelling. In the same vein, the framework is not exclusive for granular materials.
+
 # Statement of need
 
 Understanding the link from particle motions to the macroscopic material response is essential to develop accurate models for processes such as 3D printing with metal powders, pharmaceutical powder compaction, flow and handling of cereals in the alimentary industry, grinding and transport of construction materials. Discrete Element Method (DEM) has been used widely as the fundamental tool to produce the data to understand such link. However, DEM simulations are highly computationally intensive and some of the parameters used in the contact laws cannot be directly measured experimentally.
 
 GrainLearning [@Cheng2023] arises as a tool for Bayesian calibration of such computational models, which means the model parameters are estimated with a certain level of uncertainty, constrained on (noisy) real-world observations. Effectively, this makes the simulations digital twins of real-world processes with uncertainties propagated on model outputs, which then can be used for optimization or decision-making.
 
-GrainLearning started in the geotechnical engineering community and was primarily used for granular materials in quasi-static, laboratory conditions [@Cheng2018a; @Cheng2019]. These include triaxial [@Hartmann2022; @LI2024105957] and oedometric [@Cheng2019] compressions of soil samples.
-In the particle technology community, attempts with GrainLearning have been made to identify contact parameters for polymer and pharmaceutical powders against angle-of-repose [@essay91991], shear cell [@Thornton2023], and sintering experiments [@ALVAREZ2022117000]. Satisfactory results have been obtained in simulation cases where the grains were in dynamic regimes or treated under multi-physical processes.
-
-# State of the field
-
 Conventionally, the calibration of contact parameters at the grain scale is accomplished by trial and error, by comparing the macroscopic responses between simulation and experiments. This is due to the difficulty of obtaining precise measurements at the contact level and the randomness of grain properties (e.g., shape, stiffness, and asphericity).
 In the last decade, optimization [@Do2018] and design-of-experiment [@Hanley2011] approaches such as Latin Hypercube sampling and genetic algorithms have been used. However, the amount of model runs is still too large.
 For this reason, Gaussian process regression [@Fransen2021] or artificial neural networks [@Benvenuti2016] were tested as surrogate- or meta-models for the DEM.
 GrainLearning combines probabilistic learning of parameter space and sampling to achieve Bayesian optimization efficiently.
 
 # Functionality
 
-- **Calibration**: By means of Sequential Monte Carlo filtering GrainLearning can infer and update model parameters. By learning the underlying distribution using a variational Gaussian model, highly probable zones are identified and sampled iteratively until a tolerance for the overall uncertainty is reached. This process requires the input of: a time series reference data, the ranges of the parameters to infer and a tolerance. The software iteratively minimizes the discrepancy between the model solution and the reference data.
+GrainLearning started in the geotechnical engineering community and was primarily used for granular materials in quasi-static, laboratory conditions [@Cheng2018a; @Cheng2019]. These include triaxial [@Hartmann2022; @LI2024105957] and oedometric [@Cheng2019] compressions of soil samples.
+In the particle technology community, attempts with GrainLearning have been made to identify contact parameters for polymer and pharmaceutical powders against angle-of-repose [@essay91991], shear cell [@Thornton2023], and sintering experiments [@ALVAREZ2022117000]. Satisfactory results have been obtained in simulation cases where the grains were in dynamic regimes or treated under multi-physical processes.
+
+- **Calibration or parameter inference**: By means of Sequential Monte Carlo filtering GrainLearning can infer and update model parameters. By learning the underlying distribution using a variational Gaussian model, highly probable zones are identified and sampled iteratively until a tolerance for the overall uncertainty is reached. This process requires the input of: a time series reference data, the ranges of the parameters to infer and a tolerance. The software iteratively minimizes the discrepancy between the model solution and the reference data.
+
+![Elements of the trade in the calibration process. 1. Draw initial values of the parameters to calibrate. 2. Run the dynamic system with the parameters. 3. With the reference data or observation, estimate the posterior distribution via the bayesian filtering. 4. Check convergence of the parameter inference, if the process has not converged: 5. Define a gaussian mixture from the examples of this iteration and sample the parameters for the next iteration. 6. Next iteration step.
+For more details check [the iterative bayesian filter section of GrainLearning's documentation].(https://grainlearning.readthedocs.io/en/latest/bayesian_filtering.html#iterative-bayesian-filter)
+\label{fig:calibration_diagram}](calibration_diagram.png)
+
 - **Surrogate modeling**: Besides using direct simulation results (e.g. DEM) GrainLearning offers the capability of building surrogates (e.g. recurrent neural networks) as an alternative to computationally expensive DEM simulations, effectively reducing the cost by several orders of magnitude.
 
 # Acknowledgements