Equivariant deep neural networks under finite group for solving partial differential equations

Incorporating properties of partial differential equations (PDEs) or their solutions into the loss function of physics-informed neural networks (PINN) is a common strategy to improve its performance, yet it frequently introduces difficulties associated with multi-objective optimization. In this paper, we embed the finite group of either PDEs or their solutions directly into the network architecture and construct an equivariant deep neural network (eDNN) for solving PDEs, whose architecture is inherently equivariant under the corresponding finite group. Specifically, we propose a general framework for the eDNN under the finite groups with linear transformations on the dependent variables, and also design an explicit network architecture for a nonlinear finite group. As a result, the proposed eDNN exhibits enhanced learning performance and extrapolation capability. Numerical experiments verify that the eDNN outperforms the vanilla PINN and its certain variants in prediction accuracy within and beyond the training domain with fewer sampling points, while preserving the underlying finite group properties.