Novel and general discontinuity-removing PINNs for elliptic interface problems

Abstract

This paper proposes a novel and general framework of the discontinuity-removing physics-informed neural networks (DR-PINNs) for addressing elliptic interface problems. In the DR-PINNs, the solution is split into a smooth component and a non-smooth component, each represented by a separate network surrogate that can be trained either independently or together. The decoupling strategy involves training the two components sequentially. The first network handles the non-smooth part and pre-learns partial or full jumps to assist the second network in learning the complementary PDE conditions. Three decoupling strategies of handling interface problems are built by removing some jumps and incorporating cusp-capturing techniques. On the other hand, the decoupled approaches rely heavily on the cusp-enforced level-set function and are less efficient due to the need for two separate training stages. To overcome these limitations, a novel DR-PINN coupled approach is proposed in this work, where both components learn complementary conditions simultaneously in an integrated single network, eliminating the need for cusp-enforced level-set functions. Furthermore, the stability and accuracy of training are enhanced by an innovative architecture of the lightweight feedforward neural network (FNN) and a powerful geodesic acceleration Levenberg-Marquardt (gd-LM) optimizer. Several numerical experiments illustrate the effectiveness and great potential of the proposed method, with accuracy outperforming most deep neural network approaches and achieving the state-of-the-art results.