Variational Physics-Informed Neural Networks For Solving Partial Differential Equations
This work addresses the computational efficiency and accuracy issues in solving PDEs for researchers in scientific computing and machine learning, representing an incremental improvement over existing PINN methods.
The authors tackled the challenge of solving partial differential equations (PDEs) by developing a variational physics-informed neural network (VPINN) that uses a Petrov-Galerkin approach with Legendre polynomials as the test space, resulting in reduced training cost and increased accuracy compared to standard PINNs, as demonstrated in examples showing clear advantages in both speed and accuracy.
Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a Petrov-Galerkin version of PINNs based on the nonlinear approximation of deep neural networks (DNNs) by selecting the {\em trial space} to be the space of neural networks and the {\em test space} to be the space of Legendre polynomials. We formulate the \textit{variational residual} of the PDE using the DNN approximation by incorporating the variational form of the problem into the loss function of the network and construct a \textit{variational physics-informed neural network} (VPINN). By integrating by parts the integrand in the variational form, we lower the order of the differential operators represented by the neural networks, hence effectively reducing the training cost in VPINNs while increasing their accuracy compared to PINNs that essentially employ delta test functions. For shallow networks with one hidden layer, we analytically obtain explicit forms of the \textit{variational residual}. We demonstrate the performance of the new formulation for several examples that show clear advantages of VPINNs over PINNs in terms of both accuracy and speed.