BINet: Learn to Solve Partial Differential Equations with Boundary Integral Networks

Abstract

We propose a method combining boundary integral equations and neural networks (BINet) to solve (parametric) partial differential equations (PDEs) and operator problems in both bounded and unbounded domains. For PDEs with explicit fundamental solutions, BINet learns to solve, as a proxy, associated boundary integral equations using neural networks. The benefits are three-fold. Firstly, only the boundary conditions need to be fitted since the PDE can be automatically satisfied with single or double layer potential according to the potential theory. Secondly, the dimension of the boundary integral equations is less by one, and as such, the sample complexity can be reduced significantly. Lastly, in the proposed method, all differential operators have been removed, hence the numerical efficiency and stability are improved. Adopting neural tangent kernel (NTK) techniques, we provide proof of the convergence of BINets in the limit that the width of the neural network goes to infinity. Extensive numerical experiments show that, without calculating high-order derivatives, BINet is much easier to train and usually gives more accurate solutions, especially in the cases that the boundary conditions are not smooth enough. Further, BINet outperforms strong baselines for both one single PDE and parameterized PDEs in the bounded and unbounded domains.

Keywords

Partial differential equation / boundary integral / neural network / learning operator