A Domain Decomposition Deep Neural Network Method with Multi-Activation Functions for Solving Elliptic and Parabolic Interface Problems

TL;DR

This paper introduces a domain decomposition deep neural network method with multi-activation functions for efficiently solving high-dimensional elliptic and parabolic interface problems with discontinuous coefficients, improving accuracy near interfaces.

Contribution

The paper proposes a novel multi-activation function approach within a domain decomposition neural network framework, enhancing learning near interfaces in high-dimensional PDE problems.

Findings

Effective in 2D to 10D problems with complex interfaces

Conditional error bounds relate solution accuracy to loss and quadrature errors

Numerical results validate improved accuracy and efficiency

Abstract

We present a domain decomposition-based deep learning method for solving elliptic and parabolic interface problems with discontinuous coefficients in two to ten dimensions. Our Multi-Activation Function (MAF) approach employs two independent neural networks, one for each subdomain, coupled through interface conditions in the loss function. The key innovation is a multi-activation mechanism within each subdomain network that adaptively blends multiple activation functions (e.g., $\tanh$ and Gaussian-type) with interface-aware weighting, enhancing learning efficiency near interfaces where coupling constraints are most demanding. We prove conditional error bounds relating solution accuracy to trained loss values and quadrature errors. Numerical experiments on elliptic and parabolic interface problems with various interface geometries (2D--10D) validate the effectiveness and accuracy of the proposed method.