Symmetry group based domain decomposition to enhance physics-informed neural networks for solving partial differential equations

TL;DR

The proposed method can predict high-accuracy solutions of PDEs which are failed by the vanilla PINN in the whole domain and the extended physics-informed neural network in the same sub-domains.

Abstract

Domain decomposition provides an effective way to tackle the dilemma of physics-informed neural networks (PINN) which struggle to accurately and efficiently solve partial differential equations (PDEs) in the whole domain, but the lack of efficient tools for dealing with the interfaces between two adjacent sub-domains heavily hinders the training effects, even leads to the discontinuity of the learned solutions. In this paper, we propose a symmetry group based domain decomposition strategy to enhance the PINN for solving the forward and inverse problems of the PDEs possessing a Lie symmetry group. Specifically, for the forward problem, we first deploy the symmetry group to generate the dividing-lines having known solution information which can be adjusted flexibly and are used to divide the whole training domain into a finite number of non-overlapping sub-domains, then utilize the PINN and the symmetry-enhanced PINN methods to learn the solutions in each sub-domain and finally stitch them to the overall solution of PDEs. For the inverse problem, we first utilize the symmetry group acting on the data of the initial and boundary conditions to generate labeled data in the interior domain of PDEs and then find the undetermined parameters as well as the solution by only training the neural networks in a sub-domain. Consequently, the proposed method can predict high-accuracy solutions of PDEs which are failed by the vanilla PINN in the whole domain and the extended physics-informed neural network in the same sub-domains. Numerical results of the Korteweg-de Vries equation with a translation symmetry and the nonlinear viscous fluid equation with a scaling symmetry show that the accuracies of the learned solutions are improved largely.

Figures (13)

• Figure 1: (Color online) KdV equation: (A) Distribution of training data for the PINN. (B) $L_2$ relative errors by the PINN with respect to different $b$.
• Figure 2: (Color online) KdV equation: comparisons of exact solution and predicted solution and absolute errors of PINN. (A) Exact solution. (B) Predicted solution. (C) Absolute errors.
• Figure 3: (Color online): Schematic diagram of the sdPINN method.
• Figure 4: (Color online) KdV equation: Distributions of training data for the XPINN and sdPINN. The two methods take the same number of training data but do not use the same number of training points due to the random sampling.
• Figure 5: (Color online) KdV equation: Comparisons of exact solution and predicted solution and absolute errors of XPINN, sdPINN and sdPINN-isc. (A) Exact solution. (B) Predicted solution by XPINN. (C) Predicted solution by sdPINN. (D) Predicted solution by sdPINN-isc. (E) Absolute error by XPINN. (F) Absolute error by sdPINN. (G) Absolute error by sdPINN-isc.