Solving 2-D Helmholtz equation in the rectangular, circular, and elliptical domains using neural networks

TL;DR

This work addresses solving the 2-D Helmholtz equation with complex boundary geometries using physics-informed neural networks. It identifies vanishing-gradient issues when using Lagrange multipliers and introduces a trial-solution method based on transfinite interpolation and $R$-functions to enforce boundary conditions a priori, enabling unconstrained interior optimization across rectangular, circular, and elliptical domains. The approach yields acoustic-field predictions that closely match 2-D FEM results across a range of frequencies, including higher-frequency regimes where standard PINN formulations struggle. The method leverages boundary-geometry-inspired distance functions and an interior residual loss, with careful handling of corner discontinuities, offering a pathway toward domain-agnostic neural solvers for Helmholtz-type problems.

Abstract

Physics-informed neural networks offered an alternate way to solve several differential equations that govern complicated physics. However, their success in predicting the acoustic field is limited by the vanishing-gradient problem that occurs when solving the Helmholtz equation. In this paper, a formulation is presented that addresses this difficulty. The problem of solving the two-dimensional Helmholtz equation with the prescribed boundary conditions is posed as an unconstrained optimization problem using trial solution method. According to this method, a trial neural network that satisfies the given boundary conditions prior to the training process is constructed using the technique of transfinite interpolation and the theory of R-functions. This ansatz is initially applied to the rectangular domain and later extended to the circular and elliptical domains. The acoustic field predicted from the proposed formulation is compared with that obtained from the two-dimensional finite element methods. Good agreement is observed in all three domains considered. Minor limitations associated with the proposed formulation and their remedies are also discussed.

Figures (26)

• Figure 1: Schematic diagram of a feedforward neural network.
• Figure 1: Schematic diagram of a rectangular domain with discretization.
• Figure 2: Schematic diagram of the rectangular domain with the boundary conditions.
• Figure 3: Schematic diagram of a feedforward neural network for the 2-D domain.
• Figure 4: Acoustic field distribution at 300 Hz, 600 Hz, and 750 Hz: (a, b, c) from FEM, (d, e, f) from ANN ($\mathcal{L} = \mathcal{L}_d + \mathcal{L}_b$), and (g, h, i) are the errors (FEM-ANN).