IG-PINNs: Interface-gated physics-informed neural networks for solving elliptic interface problems

TL;DR

The paper tackles elliptic interface problems with discontinuities by introducing IG-PINNs, which combine a global fully connected network for smooth domain behavior with interface-gated networks that correct at the interface. A gating mechanism and a level-set-based transmitter carry interface information to specialized IG-NNs within each subdomain, while a shared network learns the overall solution. Extensive 2D/3D experiments demonstrate that IG-PINNs achieve higher accuracy than PINNs, I-PINNs, and M-PINNs, particularly in jump and flux constraints, with ablations confirming the critical role of the interface modules. Although this approach incurs higher computational cost, it provides a principled and effective framework for handling complex and high-dimensional interface problems, suggesting avenues for efficiency-focused improvements.

Abstract

In this work, we develop interface-gated physics-informed neural networks (IG-PINNs) to solve elliptic interface equations. In IG-PINNs, we use a fully connected neural network to capture the smooth behavior across the entire domain. In each subdomain separated by the interface, an interface-gated network is utilized to provide corrections at the interface. In the architectural design of the interface-gated network, we introduce a gating mechanism and a level-set function derived from the interface. This design enables the interface-gated network to effectively handle discontinuous jumps across the interface. Some numerical experiments have confirmed the effectiveness of the IG-PINNs, demonstrating higher accuracy compared with PINNs, interface PINNs (I-PINNs) and multi-domain PINNs (M-PINNs).

Figures (17)

• Figure 1: (a): interface-gated module, where the gray rectangle represents a fully connected neural network layer $Z^{n}$. (b): interface-gated neural network (IG-NNs). (c): Interface-gated PINNs (IG-PINNs).
• Figure 2: Ablation experiment, (a): Absolute errors of IG-PINNs. (b): Absolute errors without IG-NNs. (c): Absolute errors without $\mu_{nn}(x,y)$. (d): Absolute errors with simplified gating (without Q, K, V). (e): Absolute errors for a single network with equivalent parameters.
• Figure 3: Example \ref{['E1']}, (a): Absolute errors of M-PINNs, I-PINNs and IG-PINNs. (b): IG-PINNs prediction, exact solution. (c): PINNs absolute errors. (d): PINNs prediction, exact solution.
• Figure 4: Example \ref{['E1']}, network components of IG-PINNs
• Figure 5: Example \ref{['NN-FEM']}, (a): IG-PINNs absolute errors. (b): M-PINNs absolute errors. (c): I-PINNs absolute errors. (d): PINNs absolute errors.

Theorems & Definitions (6)

• Example 3.1
• Example 3.2
• Example 3.3
• Example 3.4
• Example 3.5
• Example 3.6