BINet: Learn to Solve Partial Differential Equations with Boundary Integral Networks (CSIAM Transactions on Applied Mathematics, 2023)
We propose a method combining boundary integral equations and neuralnetworks (BINet) to solve (parametric) partial differential equations (PDEs) and operator problems in both bounded and unbounded domains. For PDEs with explicitfundamental solutions, BINet learns to solve, as a proxy, associated boundary integral equations using neural networks. The benefits are three-fold. Firstly, only theboundary conditions need to be fitted since the PDE can be automatically satisfiedwith single or double layer potential according to the potential theory. Secondly, thedimension of the boundary integral equations is less by one, and as such, the samplecomplexity can be reduced significantly. Lastly, in the proposed method, all differential operators have been removed, hence the numerical efficiency and stability areimproved. Adopting neural tangent kernel (NTK) techniques, we provide proof ofthe convergence of BINets in the limit that the width of the neural network goes toinfinity. Extensive numerical experiments show that, without calculating high-orderderivatives, 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 PDEsin the bounded and unbounded domains.
Keywords: Partial differential equation; boundary integral; neural network; learning operator.