From 3c753dbe5b3516043447e29fe11029d00cd88f7e Mon Sep 17 00:00:00 2001 From: Thivin Anandh <52538701+thivinanandh@users.noreply.github.com> Date: Tue, 30 Jul 2024 00:24:21 +0530 Subject: [PATCH] Joss paper Changes - 4 (#21) Squash and merge - joss_paper to main branch --- joss/jats/paper.jats | 46 +++++++++++++++++++++++-------------------- joss/paper.md | 21 ++++++++++---------- joss/paper.pdf | Bin 317564 -> 317856 bytes 3 files changed, 36 insertions(+), 31 deletions(-) diff --git a/joss/jats/paper.jats b/joss/jats/paper.jats index 402daa4..4fce842 100644 --- a/joss/jats/paper.jats +++ b/joss/jats/paper.jats @@ -151,9 +151,9 @@ a Creative Commons Attribution 4.0 International License (CC BY neurons, the mathematical representation of the function takes the following shape: - - uNN(x;W,b)=lT(h)T(h1)T1(x)

+ uNN(x;W,b)=lT(h)T(h1)T1(x).

Here, l:nh @@ -191,12 +191,12 @@ a Creative Commons Attribution 4.0 International License (CC BY

- 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.

+ 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, @@ -271,17 +271,17 @@ a Creative Commons Attribution 4.0 International License (CC BY

The loss function of hp-VPINNs with 𝙽_𝚎𝚕𝚎𝚖 - elements in the domain can be written as follows. + 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 + Lv(W,b)=1𝙽_𝚎𝚕𝚎𝚖k=1𝙽_𝚎𝚕𝚎𝚖(KkuNN(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 @@ -303,7 +303,7 @@ a Creative Commons Attribution 4.0 International License (CC BY (Anandh et al., 2024) and (Ganesan - & Tobiska, 2017)

+ & Tobiska, 2017).

Statement of need @@ -323,15 +323,19 @@ a Creative Commons Attribution 4.0 International License (CC BY 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 +

In this work, we present the python based implementation of the + novel FastVPINNs framework which 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.

+ et al., 2015). FastVPINNs 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. As per the authors’ + knowledge, FastVPINNs is the first open-source implementation of the + hp-VPINNs framework with tensor-based loss computation and support for + complex geometries.

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, @@ -394,7 +398,7 @@ a Creative Commons Attribution 4.0 International License (CC BY 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

+ an hp-VPINNs solver.

Data Module: @@ -428,7 +432,7 @@ a Creative Commons Attribution 4.0 International License (CC BY

With the higher level of abstraction provided by the FastVPINNs framework, users can solve a PDE with just six API calls. A Minimal working example to solve the Poisson equation using the FastVPINNs - framework ca be found + framework can be found here. The example files with detailed documentation can be found in the Tutorials diff --git a/joss/paper.md b/joss/paper.md index 3882a12..79c1908 100644 --- a/joss/paper.md +++ b/joss/paper.md @@ -39,7 +39,7 @@ Here, $x \in \Omega$ is the spatial co-ordinate, $u(x)$ is the solution of the P 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) + 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*} @@ -49,8 +49,8 @@ Physics-informed neural networks (PINNs), introduced by [@raissi2019physics], wo \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_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*} @@ -68,23 +68,24 @@ Variational Physics informed neural networks (VPINNs) [@kharazmi2019variational] \end{equation*} -The loss function of hp-VPINNs with $\texttt{N\_{elem}}$ elements in the domain can be written as follows. +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_v(W,b) &= \frac{1}{\texttt{N\_elem}}\sum_{k=1}^{\texttt{N\_elem}} \left( \int_{K_k} \nabla u_{\text{NN}}(x;W,b) \cdot \nabla v_k dK ~ - ~ \int_{K_k} f\,v_k\,dK ~\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] +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. +In this work, we present the python based implementation of the novel FastVPINNs framework which is built using TensorFlow-v2.0 [@tensorflow2015-whitepaper]. FastVPINNs 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. As per the authors' knowledge, FastVPINNs is the first open-source implementation of the hp-VPINNs framework with tensor-based loss computation and support for complex geometries. + 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. @@ -107,7 +108,7 @@ The FE module is responsible for handling the finite element test functions and \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* +*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. @@ -120,7 +121,7 @@ This module contains custom subclasses of the `tensorflow.keras.Model` class, wh # 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. A Minimal working example to solve the Poisson equation using the FastVPINNs framework ca be found [here](https://cmgcds.github.io/fastvpinns/#usage). The example files with detailed documentation can be found in the [Tutorials section](https://cmgcds.github.io/fastvpinns/_rst/_tutorial.html) of the documentation. +With the higher level of abstraction provided by the FastVPINNs framework, users can solve a PDE with just six API calls. A Minimal working example to solve the Poisson equation using the FastVPINNs framework can be found [here](https://cmgcds.github.io/fastvpinns/#usage). The example files with detailed documentation can be found in the [Tutorials section](https://cmgcds.github.io/fastvpinns/_rst/_tutorial.html) of the documentation. # Testing diff --git a/joss/paper.pdf b/joss/paper.pdf index 500acb1d84919e5780c9880d567dbfe02b2b33ce..9c72a4bd4ecefe68debc91646736b44d43dd78f7 100644 GIT binary patch delta 31943 zcmV)7K*zuQuoIxO6R;r*m+=<_6@TqL%Z?)Wgw*n{Cy zFT6f<+ZbQw_m|X@Qam%tXxKnux+XJ}dXN-JQ6#0{6Nmr)`G4_)PyhXg7ZJaRNm#*@ z(32E`nINZcKVE)&VU9Z=Iw4?8B0r29+h|3U`%^7ZAHhksCZ652B_ ze3-uebrqnmKc+vwaxM+$O2MauKL_8<+h4x^@rBP2yMNun?N?}lmH)8Nvcaqslh#kh zcxI)~GzlSS9jlP9@h4$yd@zAITZ(`AuODK%{R2ka8uLg=gUd6n?R0UDGdWPwF)0iH z)lt$iZ}nsYOo|hglI~4<_p@Y@ys9_bH%6JOpgTI<(Dcl~&~D5bEi z<-tK64R!Q2KEtorSE+%$)>0gWr%?^eA-_YpY7oSFwI#w~uDOSRExVC#c|_Iv)3SXx z*A>3_jj}j*%X9HWpi8Eu_57GmvW~IwNEi0-NWj!FH0iJw=2%Ph0I6hdPC} zA<+prN>#oloN-;ov8M=fT+=tJfw9ofA z@<~ef^bu?Q94UdZ{C|Y1(+B4lCj%6c&~;js%*1e+jL6pSfVy=E$|Iqs@@FKBP2e%; zmzhbh1ORzNz&3v+>Ry7{)*GO4+hQ~A7=?x%;*_T}E+8V~y%)uCec4$y| 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