From 85b5b327b472d4430f3c494401a14186db821fcc Mon Sep 17 00:00:00 2001 From: Thivin Anandh <52538701+thivinanandh@users.noreply.github.com> Date: Sat, 27 Jul 2024 01:04:57 +0530 Subject: [PATCH] Joss paper Merge (#19) Squashing and merging all the changes from the joss paper branch into a single commit on main branch * Joss paper Added for the paper Added for the references Added in the github workflowa * Joss CI bug Fix FIxed the path on the joss-paper.yml * Removed Older Joss Paper folder * Added name to Joss PDF CI * Fixed broken citations * Removed the compatibility test for push to other branches * Changes to Joss Paper Changed Module name from FE2D to FE * Changed PNG Errors * Paper corrections * Paper Correction * Paper corrections - 3 * Paper corrections - 4 * Paper Correction 5 * Paper Corrections * Update paper.bib * Modifications to Joss Paper --------- Co-authored-by: Divij --- .github/workflows/compatibility-tests.yml | 2 +- .github/workflows/joss-paper.yml | 27 + joss/fastvpinns_modules.png | Bin 0 -> 111408 bytes joss/jats/fastvpinns_modules.png | Bin 0 -> 111408 bytes joss/jats/paper.jats | 649 ++++++++++++++++++++++ joss/paper.bib | 105 ++++ joss/paper.md | 189 +++++++ joss/paper.pdf | Bin 0 -> 345730 bytes joss/paper_local_compiled.pdf | Bin 0 -> 357135 bytes 9 files changed, 971 insertions(+), 1 deletion(-) create mode 100644 .github/workflows/joss-paper.yml create mode 100644 joss/fastvpinns_modules.png create mode 100644 joss/jats/fastvpinns_modules.png create mode 100644 joss/jats/paper.jats create mode 100644 joss/paper.bib create mode 100644 joss/paper.md create mode 100644 joss/paper.pdf create mode 100644 joss/paper_local_compiled.pdf diff --git a/.github/workflows/compatibility-tests.yml b/.github/workflows/compatibility-tests.yml index c88f131..c79aa07 100644 --- a/.github/workflows/compatibility-tests.yml +++ b/.github/workflows/compatibility-tests.yml @@ -2,7 +2,7 @@ name: Compatibility tests on: push: - branches: [ '*' ] + branches: [ 'main' ] pull_request: branches: [ main ] workflow_dispatch: diff --git a/.github/workflows/joss-paper.yml b/.github/workflows/joss-paper.yml new file mode 100644 index 0000000..0da7384 --- /dev/null +++ b/.github/workflows/joss-paper.yml @@ -0,0 +1,27 @@ +name: Joss Paper Compiler + +on: + push: + branches: [ "joss_paper" ] + +jobs: + paper: + runs-on: ubuntu-latest + name: Paper Draft + steps: + - name: Checkout + uses: actions/checkout@v4 + - name: Build draft PDF + uses: openjournals/openjournals-draft-action@master + with: + journal: joss + # This should be the path to the paper within your repo. + paper-path: joss/paper.md + - name: Upload + uses: actions/upload-artifact@v1 + with: + name: paper + # This is the output path where Pandoc will write the compiled + # PDF. 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+ + + + +Journal of Open Source Software +JOSS + +2475-9066 + +Open Journals + + + +0 +N/A + +FastVPINNs: An efficient tensor-based Python library for +solving partial differential equations using hp-Variational Physics +Informed Neural Networks + + + +https://orcid.org/0000-0003-4969-3242 + +Anandh +Thivin + + +* + + +https://orcid.org/0009-0005-6295-543X + +Ghose +Divij + + + + +https://orcid.org/0000-0003-1858-3972 + +Ganesan +Sashikumaar + + +* + + + +Department of Computational and Data Sciences, Indian +Institute of Science, Bangalore, India + + + + +* E-mail: +* E-mail: + + +17 +5 +2023 + +¿VOL? +¿ISSUE? +¿PAGE? + +Authors of papers retain copyright and release the +work under a Creative Commons Attribution 4.0 International License (CC +BY 4.0) +2022 +The article authors + +Authors of papers retain copyright and release the work under +a Creative Commons Attribution 4.0 International License (CC BY +4.0) + + + +python +physics-informed neural networks +scientific machine learning +partial differential equations +hp-variational physics informed neural networks +tensorflow + + + + + + Introduction +

Partial differential equations (PDEs) are essential in modeling + physical phenomena such as heat transfer, electromagnetics and fluid + dynamics. The current progress in the field of scientific machine + learning (SciML) has resulted in incorporating deep learning and + data-driven methods to solve PDEs. Let us consider a two-dimensional + Poisson equation as an example, defined on the domain + + + Ω, + which reads as follows:

+

+ + 2u(x)=f(x),Ω2,u(x)=g(x),xΩ.

+

Here, + + xΩ + is the spatial co-ordinate, + + u(x) + is the solution of the PDE, + + f(x) + is a known source term and + + g(x) + is the value of the solution on the domain boundary, + + + Ω.

+

A neural network is a parametric function of + + + x, + denoted as + + uNN(x;W,b). + In this context, + + W + and + + b + represent the weights and biases of the network. When the neural + network consists of + + h + hidden layers, with the + + ith + layer containing + + ni + neurons, the mathematical representation of the function takes the + following shape: + + + uNN(x;W,b)=lT(h)T(h1)T1(x)

+

Here, + + l:nh + is a linear mapping in the output layer and + + + T(i)()=σ(W(i)×+b(i)) + is a non-linear mapping in the + + ith + layer + + (i=1,2,,h), + with the non-linear activation function + + σ + and the weights + + W(i) + and biases + + b(i) + of the respective layers.

+

Physics-informed neural networks (PINNs), introduced by + (Raissi + et al., 2019), work by incorporating the governing equations of + the physical systems and the boundary conditions as physics-based + constrains to train the neural networks. The core idea of PINNs lies + in the ability of obtaining the gradients of the solutions with + respect to the input using the automatic differention routines + available within deep learning frameworks such as tensorflow + (Abadi + et al., 2015). The loss function for the PINNs can be written + as follows:

+

+ + Lp(W,b)=1NTt=1NT|(2uNN(xt;W,b)f(xt))|2,Lb(W,b)=1NDd=1ND|uNN(xd;W,b)g(xd)|2,LPINN(W,b)=Lp+τLb.

+

Here, the output of the neural network, + + + uNN(x;W,b), + is used to approximate the unknown solution. In addition, + + + NT + is the total number of training points in the interior of the domain + + + Ω, + + + ND + is the total number of training points on the boundary + + + Ω, + + + τ + is a scaling factor applied to control the penalty on the boundary + term and + + LPINN(W,b) + is the loss function of the PINNs.

+

Variational Physics informed neural networks (VPINNs) + (Kharazmi + et al., 2019) are an extension of PINNs, where the weak form of + the PDE is used in the loss function of the neural network. The weak + form of the PDE is obtained by multiplying the PDE with a test + function, integrating over the domain and applying integration by + parts to the higher order derivative terms. The method of hp-VPINNs + (Kharazmi + et al., 2021) was subsequently developed to increase the + accuracy using h-refinement (increasing number of elements) and + p-refinement (increasing the number of test functions). The domain + + + Ω + is divided into an array of non-overlapping cells, labeled as + + + Kk, + where + + k=1,2,,𝙽_𝚎𝚕𝚎𝚖, + ensuring that the complete union + + k=1𝙽_𝚎𝚕𝚎𝚖Kk=Ω + covers the entire domain + + Ω. + The hp-VPINNs framework utilizes specific test functions + + + vk, + where + + k + ranges from 1 to N_elem, that are localized and defined within + individual non-overlapping element across the domain. + + + vk={vp0,over Kk,0,elsewhere.

+

The loss function of hp-VPINNs with + + 𝙽_𝚎𝚕𝚎𝚖 + elements in the domain can be written as follows. + + + Lv(W,b)=1𝙽_𝚎𝚕𝚎𝚖k=1𝙽_𝚎𝚕𝚎𝚖|Kk(uNN(x;W,b)vkdKKkfvkdK)|2,Lb(W,b)=1NDd=1ND|uNN(x;W,b)g(x)|2,LVPINN(W,b)=Lv+τLb.

+

Where + + Kk + is the + + kth + element in the domain, + + vk + is the test function in the respective element. Further, + + + Lv(W,b) + is the weak form PDE residual and + + LVPINN(W,b) + is the loss function of the hp-VPINNs. For more information on the + derivation of the weak form of the PDE and the loss function of + hp-VPINNs, refer to + (Anandh + et al., 2024) and + (Ganesan + & Tobiska, 2017)

+
+ + Statement of need +

The existing implementation of hp-VPINNs framework + (Kharazmi, + 2023) suffers from two major challenges. One being the inabilty + of the framework to handle complex geometries and the other being the + increased training time associated with the increase in number of + elements within the domain. In the work + (Anandh + et al., 2024), we presented FastVPINNs, which addresses both of + these challenges. FastVPINNs handles complex geometries by using + bilinear transformation, and it uses a tensor-based loss computation + to reduce the dependency of training time on number of elements. The + current implementation of FastVPINNs can acheive an speed-up of upto a + 100 times when compared with the existing implementation of hp-VPINNs. + We have also shown that with proper hyperparameter selection, + FastVPINNs can outperform PINNs both in terms of accuracy and training + time, especially for problems with high frequency solutions.

+

Our FastVPINNs framework is built using + TensorFlow-v2.0(Abadi + et al., 2015), and provides an elegant API for users to solve + both forward and inverse problems for PDEs like the Poisson, Helmholtz + and Convection-Diffusion equations. With the current level of API + abstraction, users should be able to solve PDEs with less than six API + calls as shown in the minimal working example section. The framework + is well-documented with examples, which can enable users to get + started with the framework with ease.

+

The ability of the framework to allow users to train a hp-VPINNs to + solve a PDE both faster and with minimal code, can result in + widespread application of this method on several real-world problems, + which often require complex geometries with a large number of elements + within the domain.

+
+ + Modules +

The FastVPINNs framework consists of five core modules, which are + Geometry, FE, Data, Physics and Model.

+ +

FastVPINNs Modules

+ +
+ + Geometry Module +

This module provides the functionality to define the geometry of + the domain. The user can either generate a quadrilateral mesh + internally or read an external .mesh file. + The module also provides the functionality to obtain boundary points + for complex geometries.

+
+ + FE Module +

The FE module is responsible for handling the finite element test + functions and their gradients. The module’s functionality can be + broadly classified into into four categories:

+ + +

Test functions: Provides the test function + values and its gradients for a given reference element. In our + framework, we use Lagrange polynomials as the test + functions.

+
+ +

Quadrature functions: Provides the quadrature + points and weights for a given element based on the quadrature + order and the quadrature method selected by the user. This + method will be used for performing the numerical integration of + the weak form residual of the PDE.

+
+ +

Transformation functions: Provides the + implementation of transformation routines such as affine + transformation and bilinear transformation. This can be used to + transform the test function values and gradients from the + reference element to the actual element.

+
+ +

Finite Element Setup: Provides the functionality + to set up the test functions, quadrature rules and the + transformation for every element and save them in a common class + to access them. Further, it also hosts functions to plot the + mesh with quadrature points, assign boundary values based on the + boundary points obtained from the geometry module and calculate + the forcing term in the residual.

+
+
+

Remark: The module is named FE (Finite Element) Module + because of its similarities with classical FEM routines, such as + test functions, numerical quadratures and transformations. However, + we would like to state that our framework is not an FEM solver, but + an hp-VPINNs solver

+
+ + Data Module: +

The Data module collects data from all modules which are required + for training and converts them to a tensor data type with the user + specified precision (for example, tf.float32 + or tf.float64). It also assembles the test + function values and gradients to form a three-dimensional tensor, + which will be used during the loss computation.

+
+ + Physics Module: +

This module contains functions which are used to compute the + variational loss function for different PDEs. These functions accept + the test function tensors and the predicted gradients from the + neural network along with the forcing matrix to compute the PDE + residuals using tensor-based operations.

+
+ + Model Module: +

This module contains custom subclasses of the + tensorflow.keras.Model class, which are used + to train the neural network. The module provides the functionality + to train the neural network using the tensor based loss computation + and also provides the functionality to predict the solution of the + PDE for a given input.

+
+
+ + Minimal Working Example +

With the higher level of abstraction provided by the FastVPINNs + framework, users can solve a PDE with just six API calls. Shown below + is a minimal working example to solve the Poisson equation using the + FastVPINNs framework.

+ #load the geometry +domain = Geometry_2D("quadrilateral", "internal", + 100, 100, "./") +cells, boundary_points = + domain.generate_quad_mesh_internal(\ + x_limits=[0, 1], y_limits=[0, 1],\ + n_cells_x=4, n_cells_y=4,\ + num_boundary_points=400) + +# Note: Users need to provide the necessary +# boundary values and the forcing function + +# load the FEspace +fespace = Fespace2D(domain.mesh,cells,boundary_points,\ + domain.mesh_type,fe_order=5,\ + fe_type="jacobi",quad_order=5,\ + quad_type="legendre", \ + fe_transformation_type="bilinear"\ + bound_function_dict=bound_function_dict,\ + bound_condition_dict=bound_condition_dict,\ + forcing_function=rhs,\ + output_path=i_output_path,\ + generate_mesh_plot=True) + +# Instantiate Data handler +dh = datahandler2D(fespace, domain, dtype=tf.float32) + +# Instantiate the model with the loss function for the model +model = DenseModel(layer_dims=[2, 30, 30, 30, 1],\ + learning_rate_dict=0.01,params_dict=params_dict, + ## Loss function of poisson2D + loss_function=pde_loss_poisson, + input_tensors_list=\ + [in_tensor, + dir_in, + dir_out], + orig_factor_matrices=\ + [dh.shape_val_mat_list, + dh.grad_x_mat_list, + dh.grad_y_mat_list], + force_function_list=dh.forcing_function_list,\ + tensor_dtype=tf.float32, + use_attention=i_use_attention, + activation=i_activation, + hessian=False) + +# Train the model +for epoch in range(1000): + model.train_step() +
+ + Testing +

The FastVPINNs framework has a strong testing suite, which tests + the core functionalities of the framework. The testing suite is built + using the pytest framework and is integrated + with the continuous integration pipeline provided by Github Actions. + The testing functionalites can be broadly classified into the three + categories:

+ + +

Unit Testing: Covers the testing of individual + modules and their functionalities.

+
+ +

Integration Testing: Covers the overall flow of + the framework. Different PDEs are solved with different parameters + to check if the accuracy after training is within the acceptable + limits. This ensures that the collection of modules work together + as expected.

+
+ +

Compatibility Testing: The framework is tested + with different versions of Python such as 3.8, 3.9, 3.10, 3.11 and + on different versions of OS such as Ubuntu, MacOS and Windows to + ensure the compatibility of the framework across different + platforms.

+
+
+
+ + Acknowledgements +

We thank Shell Research, India, for providing partial funding for + this project. We are thankful to the MHRD Grant No.STARS-1/388 (SPADE) + for partial support.

+
+ + + + + + + RaissiMaziar + PerdikarisParis + KarniadakisGeorge E + + Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations + Journal of Computational physics + Elsevier + 2019 + 378 + 686 + 707 + + + + + + KharazmiEhsan + + hp-VPINNs: High-performance variational physics-informed neural networks + 2023 + https://github.com/ehsankharazmi/hp-VPINNs + + + + + + AnandhThivin + GhoseDivij + JainHimanshu + GanesanSashikumaar + + FastVPINNs: Tensor-driven acceleration of VPINNs for complex geometries + 2024 + https://arxiv.org/abs/2404.12063 + + + + + + KharazmiEhsan + ZhangZhongqiang + KarniadakisGeorge Em + + hp-VPINNs: Variational physics-informed neural networks with domain decomposition + Computer Methods in Applied Mechanics and Engineering + Elsevier + 2021 + 374 + 113547 + + + + + + + KharazmiEhsan + ZhangZhongqiang + KarniadakisGeorge Em + + Variational physics-informed neural networks for solving partial differential equations + arXiv preprint arXiv:1912.00873 + 2019 + + + + + + AbadiMartín + AgarwalAshish + BarhamPaul + BrevdoEugene + ChenZhifeng + CitroCraig + CorradoGreg S. + DavisAndy + DeanJeffrey + DevinMatthieu + GhemawatSanjay + GoodfellowIan + HarpAndrew + IrvingGeoffrey + IsardMichael + JiaYangqing + JozefowiczRafal + KaiserLukasz + KudlurManjunath + LevenbergJosh + ManéDandelion + MongaRajat + MooreSherry + MurrayDerek + OlahChris + SchusterMike + ShlensJonathon + SteinerBenoit + SutskeverIlya + TalwarKunal + TuckerPaul + VanhouckeVincent + VasudevanVijay + ViégasFernanda + VinyalsOriol + WardenPete + WattenbergMartin + WickeMartin + YuYuan + ZhengXiaoqiang + + TensorFlow: Large-scale machine learning on heterogeneous systems + 2015 + https://www.tensorflow.org/ + + + + + + GanesanSashikumaar + TobiskaLutz + + Finite elements: Theory and algorithms + Cambridge University Press + 2017 + + + + +
diff --git a/joss/paper.bib b/joss/paper.bib new file mode 100644 index 0000000..af6d45a --- /dev/null +++ b/joss/paper.bib @@ -0,0 +1,105 @@ +@article{raissi2019physics, + title={Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations}, + author={Raissi, Maziar and Perdikaris, Paris and Karniadakis, George E}, + journal={Journal of Computational physics}, + volume={378}, + pages={686--707}, + year={2019}, + publisher={Elsevier}, + doi = {10.1016/j.jcp.2018.10.045} +} + +@misc{hp_vpinns_github, + author = {Kharazmi, Ehsan}, + title = {{hp-VPINNs}: High-Performance Variational Physics-Informed Neural Networks}, + year = {2023}, + url = {https://github.com/ehsankharazmi/hp-VPINNs}, + note = {Accessed: 2023-12-01} +} + +@misc{anandh2024fastvpinns, + title={FastVPINNs: Tensor-Driven Acceleration + of VPINNs for Complex Geometries}, + author={Anandh, Thivin and Ghose, Divij and Jain, Himanshu + and Ganesan, Sashikumaar}, + year={2024}, + eprint={2404.12063}, + archivePrefix={arXiv}, + primaryClass={cs.LG}, + doi={10.48550/arXiv.2404.12063} +} + +@article{kharazmi2021hp, + title={{hp-VPINNs: Variational physics-informed neural networks with domain decomposition}}, + author={Kharazmi, Ehsan and Zhang, Zhongqiang and Karniadakis, George Em}, + journal={Computer Methods in Applied Mechanics and Engineering}, + volume={374}, + pages={113547}, + year={2021}, + publisher={Elsevier}, + doi={10.1016/j.cma.2020.113547} +} +@article{kharazmi2019variational, + title={{Variational physics-informed neural networks for solving partial differential equations}}, + author={Kharazmi, Ehsan and Zhang, Zhongqiang and Karniadakis, George Em}, + journal={arXiv preprint arXiv:1912.00873}, + year={2019}, + doi={10.48550/arXiv.1912.00873} +} + +@article{tensorflow2015-whitepaper, +title={ {TensorFlow}: Large-Scale Machine Learning on Heterogeneous Systems}, +url={https://www.tensorflow.org/}, +note={Software available from tensorflow.org}, +author={ + Mart\'{i}n~Abadi and + Ashish~Agarwal and + Paul~Barham and + Eugene~Brevdo and + Zhifeng~Chen and + Craig~Citro and + Greg~S.~Corrado and + Andy~Davis and + Jeffrey~Dean and + Matthieu~Devin and + Sanjay~Ghemawat and + Ian~Goodfellow and + Andrew~Harp and + Geoffrey~Irving and + Michael~Isard and + Yangqing Jia and + Rafal~Jozefowicz and + Lukasz~Kaiser and + Manjunath~Kudlur and + Josh~Levenberg and + Dandelion~Man\'{e} and + Rajat~Monga and + Sherry~Moore and + Derek~Murray and + Chris~Olah and + Mike~Schuster and + Jonathon~Shlens and + Benoit~Steiner and + Ilya~Sutskever and + Kunal~Talwar and + Paul~Tucker and + Vincent~Vanhoucke and + Vijay~Vasudevan and + Fernanda~Vi\'{e}gas and + Oriol~Vinyals and + Pete~Warden and + Martin~Wattenberg and + Martin~Wicke and + Yuan~Yu and + Xiaoqiang~Zheng}, + year={2015}, + doi={10.48550/arXiv.1603.04467} +} + +@book{ganesan2017finite, + title={Finite elements: Theory and algorithms}, + author={Ganesan, Sashikumaar and Tobiska, Lutz}, + year={2017}, + publisher={Cambridge University Press}, + doi={10.1017/9781108235013} +} diff --git a/joss/paper.md b/joss/paper.md new file mode 100644 index 0000000..b2d0ae4 --- /dev/null +++ b/joss/paper.md @@ -0,0 +1,189 @@ +--- +title: 'FastVPINNs: An efficient tensor-based Python library for solving partial differential equations using hp-Variational Physics Informed Neural Networks' +tags: + - python + - physics-informed neural networks + - scientific machine learning + - partial differential equations + - hp-variational physics informed neural networks + - tensorflow +authors: + - name: Thivin Anandh + orcid: 0000-0003-4969-3242 + affiliation: "1" + corresponding: true # corresponding author + - name: Divij Ghose + orcid: 0009-0005-6295-543X + affiliation: "1" + - name: Sashikumaar Ganesan + orcid: 0000-0003-1858-3972 + affiliation: "1" + corresponding: true # corresponding author +affiliations: + - name: Department of Computational and Data Sciences, Indian Institute of Science, Bangalore, India + index: 1 +date: 17 May 2023 +bibliography: paper.bib +--- + +# Introduction + +Partial differential equations (PDEs) are essential in modeling physical phenomena such as heat transfer, electromagnetics and fluid dynamics. The current progress in the field of scientific machine learning (SciML) has resulted in incorporating deep learning and data-driven methods to solve PDEs. Let us consider a two-dimensional Poisson equation as an example, defined on the domain $\Omega$, which reads as follows: + +\begin{align} + - \nabla^2 u(x) &= f(x), \quad \Omega \in \mathbb{R}^2, \\ + u(x) &= g(x), \quad x \in \partial \Omega. +\end{align} + +Here, $x \in \Omega$ is the spatial co-ordinate, $u(x)$ is the solution of the PDE, $f(x)$ is a known source term and $g(x)$ is the value of the solution on the domain boundary, $\partial \Omega$. + +A neural network is a parametric function of $x$, denoted as $u_{\text{NN}}(x; W, b)$. In this context, $W$ and $b$ represent the weights and biases of the network. When the neural network consists of $h$ hidden layers, with the $i^{\text{th}}$ layer containing $n_i$ neurons, the mathematical representation of the function takes the following shape: +\begin{equation*} + u_{\text{NN}}(x; W, b) = l \circ \mathrm{T}^{(h)} \circ \mathrm{T}^{(h-1)} \hdots \circ \mathrm{T}^1(x) + \label{eq:NN} +\end{equation*} + +Here, $l : \mathbb{R}^{n_h} \rightarrow \mathbb{R}$ is a linear mapping in the output layer and $\mathrm{T}^{(i)}(\cdot) = \sigma(W^{(i)} \times \cdot + b^{(i)})$ is a non-linear mapping in the $i^{th}$ layer $( i= 1, 2, \cdots , h)$, with the non-linear activation function $\sigma$ and the weights $W^{(i)}$ and biases $b^{(i)}$ of the respective layers. + +Physics-informed neural networks (PINNs), introduced by [@raissi2019physics], work by incorporating the governing equations of the physical systems and the boundary conditions as physics-based constrains to train the neural networks. The core idea of PINNs lies in the ability of obtaining the gradients of the solutions with respect to the input using the automatic differention routines available within deep learning frameworks such as tensorflow [@tensorflow2015-whitepaper]. The loss function for the PINNs can be written as follows: + +\begin{align*} + \begin{split} + L_p(W,b) &= \frac{1}{N_T}\sum_{t=1}^{N_T}\left|(-\nabla^2 u_{\text{NN}}(x_t;W,b) - f(x_t))\right|^2,\\ + L_b(W,b) &= \frac{1}{N_D}\sum_{d=1}^{N_{D}}\left|u_{\text{NN}}(x_d; W, b) - g(x_d)\right|^2,\\ + L_{\text{PINN}}(W,b) &= L_p + \tau L_b. + \end{split} +\end{align*} + +Here, the output of the neural network, $u_{\text{NN}}(x; W, b)$, is used to approximate the unknown solution. In addition, $N_T$ is the total number of training points in the interior of the domain $\Omega$, $N_D$ is the total number of training points on the boundary $\partial\Omega$, $\tau$ is a scaling factor applied to control the penalty on the boundary term and $L_{\text{PINN}}(W,b)$ is the loss function of the PINNs. + +Variational Physics informed neural networks (VPINNs) [@kharazmi2019variational] are an extension of PINNs, where the weak form of the PDE is used in the loss function of the neural network. The weak form of the PDE is obtained by multiplying the PDE with a test function, integrating over the domain and applying integration by parts to the higher order derivative terms. The method of hp-VPINNs [@kharazmi2021hp] was subsequently developed to increase the accuracy using h-refinement (increasing number of elements) and p-refinement (increasing the number of test functions). The domain $\Omega$ is divided into an array of non-overlapping cells, labeled as $K_k$, where $k=1,2,\ldots,\texttt{N\_{elem}}$, ensuring that the complete union $\bigcup_{k=1}^\texttt{N\_{elem}}K_k = \Omega$ covers the entire domain $\Omega$. The hp-VPINNs framework utilizes specific test functions $v_k$, where $k$ ranges from 1 to N\_elem, that are localized and defined within individual non-overlapping element across the domain. +\begin{equation*} + v_k= + \begin{cases} + v^p \neq 0, & \text{over $K_k$,} \\ + 0, & \text{elsewhere.} + \end{cases} + \label{eqn:test_function_summation} +\end{equation*} + + +The loss function of hp-VPINNs with $\texttt{N\_{elem}}$ elements in the domain can be written as follows. +\begin{align*} + \begin{split} + L_v(W,b) &= \frac{1}{\texttt{N\_elem}}\sum_{k=1}^{\texttt{N\_elem}}\left| \int_{K_k} \left( \nabla u_{\text{NN}}(x;W,b) \cdot \nabla v_k dK ~ - ~ \int_{K_k} f\,v_k\,dK ~\right) ~\right|^2,\\ + L_b(W,b) &= \frac{1}{N_D}\sum_{d=1}^{N_{D}}\left|u_{\text{NN}}(x; W, b) - g(x)\right|^2,\\ + L_{\text{VPINN}}(W,b) &= L_v + \tau L_b. + \end{split} +\end{align*} + +Where $K_k$ is the $k^{th}$ element in the domain, $v_k$ is the test function in the respective element. Further, $L_v(W,b)$ is the weak form PDE residual and $L_{\text{VPINN}}(W,b)$ is the loss function of the hp-VPINNs. For more information on the derivation of the weak form of the PDE and the loss function of hp-VPINNs, refer to [@anandh2024fastvpinns] and [@ganesan2017finite] + + +# Statement of need + +The existing implementation of hp-VPINNs framework [@hp_vpinns_github] suffers from two major challenges. One being the inabilty of the framework to handle complex geometries and the other being the increased training time associated with the increase in number of elements within the domain. In the work [@anandh2024fastvpinns], we presented FastVPINNs, which addresses both of these challenges. FastVPINNs handles complex geometries by using bilinear transformation, and it uses a tensor-based loss computation to reduce the dependency of training time on number of elements. The current implementation of FastVPINNs can acheive an speed-up of upto a 100 times when compared with the existing implementation of hp-VPINNs. We have also shown that with proper hyperparameter selection, FastVPINNs can outperform PINNs both in terms of accuracy and training time, especially for problems with high frequency solutions. + +Our FastVPINNs framework is built using TensorFlow-v2.0[@tensorflow2015-whitepaper], and provides an elegant API for users to solve both forward and inverse problems for PDEs like the Poisson, Helmholtz and Convection-Diffusion equations. With the current level of API abstraction, users should be able to solve PDEs with less than six API calls as shown in the minimal working example section. The framework is well-documented with examples, which can enable users to get started with the framework with ease. + +The ability of the framework to allow users to train a hp-VPINNs to solve a PDE both faster and with minimal code, can result in widespread application of this method on several real-world problems, which often require complex geometries with a large number of elements within the domain. + +# Modules + +The FastVPINNs framework consists of five core modules, which are Geometry, FE, Data, Physics and Model. + +![FastVPINNs Modules](fastvpinns_modules.png){ height=50% } + +## Geometry Module +This module provides the functionality to define the geometry of the domain. The user can either generate a quadrilateral mesh internally or read an external \verb|.mesh| file. The module also provides the functionality to obtain boundary points for complex geometries. + +## FE Module +The FE module is responsible for handling the finite element test functions and their gradients. The module's functionality can be broadly classified into into four categories: + +\begin{itemize} + \item \textbf{Test functions}: Provides the test function values and its gradients for a given reference element. In our framework, we use Lagrange polynomials as the test functions. + \item \textbf{Quadrature functions}: Provides the quadrature points and weights for a given element based on the quadrature order and the quadrature method selected by the user. This method will be used for performing the numerical integration of the weak form residual of the PDE. + \item \textbf{Transformation functions}: Provides the implementation of transformation routines such as affine transformation and bilinear transformation. This can be used to transform the test function values and gradients from the reference element to the actual element. + \item \textbf{Finite Element Setup}: Provides the functionality to set up the test functions, quadrature rules and the transformation for every element and save them in a common class to access them. Further, it also hosts functions to plot the mesh with quadrature points, assign boundary values based on the boundary points obtained from the geometry module and calculate the forcing term in the residual. +\end{itemize} + +*Remark: The module is named FE (Finite Element) Module because of its similarities with classical FEM routines, such as test functions, numerical quadratures and transformations. However, we would like to state that our framework is not an FEM solver, but an hp-VPINNs solver* + +## Data Module: +The Data module collects data from all modules which are required for training and converts them to a tensor data type with the user specified precision (for example, `tf.float32` or `tf.float64`). It also assembles the test function values and gradients to form a three-dimensional tensor, which will be used during the loss computation. + +## Physics Module: +This module contains functions which are used to compute the variational loss function for different PDEs. These functions accept the test function tensors and the predicted gradients from the neural network along with the forcing matrix to compute the PDE residuals using tensor-based operations. + +## Model Module: +This module contains custom subclasses of the `tensorflow.keras.Model` class, which are used to train the neural network. The module provides the functionality to train the neural network using the tensor based loss computation and also provides the functionality to predict the solution of the PDE for a given input. + +# Minimal Working Example {#sec:minimal-working-example} + +With the higher level of abstraction provided by the FastVPINNs framework, users can solve a PDE with just six API calls. Shown below is a minimal working example to solve the Poisson equation using the FastVPINNs framework. + +```python +#load the geometry +domain = Geometry_2D("quadrilateral", "internal", + 100, 100, "./") +cells, boundary_points = + domain.generate_quad_mesh_internal(\ + x_limits=[0, 1], y_limits=[0, 1],\ + n_cells_x=4, n_cells_y=4,\ + num_boundary_points=400) + +# Note: Users need to provide the necessary +# boundary values and the forcing function + +# load the FEspace +fespace = Fespace2D(domain.mesh,cells,boundary_points,\ + domain.mesh_type,fe_order=5,\ + fe_type="jacobi",quad_order=5,\ + quad_type="legendre", \ + fe_transformation_type="bilinear"\ + bound_function_dict=bound_function_dict,\ + bound_condition_dict=bound_condition_dict,\ + forcing_function=rhs,\ + output_path=i_output_path,\ + generate_mesh_plot=True) + +# Instantiate Data handler +dh = datahandler2D(fespace, domain, dtype=tf.float32) + +# Instantiate the model with the loss function for the model +model = DenseModel(layer_dims=[2, 30, 30, 30, 1],\ + learning_rate_dict=0.01,params_dict=params_dict, + ## Loss function of poisson2D + loss_function=pde_loss_poisson, + input_tensors_list=\ + [in_tensor, + dir_in, + dir_out], + orig_factor_matrices=\ + [dh.shape_val_mat_list, + dh.grad_x_mat_list, + dh.grad_y_mat_list], + force_function_list=dh.forcing_function_list,\ + tensor_dtype=tf.float32, + use_attention=i_use_attention, + activation=i_activation, + hessian=False) + +# Train the model +for epoch in range(1000): + model.train_step() +``` + +# Testing + +The FastVPINNs framework has a strong testing suite, which tests the core functionalities of the framework. The testing suite is built using the `pytest` framework and is integrated with the continuous integration pipeline provided by Github Actions. The testing functionalites can be broadly classified into the three categories: +\begin{itemize} + \item \textbf{Unit Testing}: Covers the testing of individual modules and their functionalities. + \item \textbf{Integration Testing}: Covers the overall flow of the framework. Different PDEs are solved with different parameters to check if the accuracy after training is within the acceptable limits. This ensures that the collection of modules work together as expected. + \item \textbf{Compatibility Testing}: The framework is tested with different versions of Python such as 3.8, 3.9, 3.10, 3.11 and on different versions of OS such as Ubuntu, MacOS and Windows to ensure the compatibility of the framework across different platforms. +\end{itemize} + +# Acknowledgements +We thank Shell Research, India, for providing partial funding for this project. 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