用神经网络求解复杂形状下的声场问题，精度媲美有限元法。
Solving 2-D Helmholtz equation in the rectangular, circular, and elliptical domains using neural networks
- 构建满足边界条件的神经网络初值解，避免梯度消失问题。
- 在矩形、圆形和椭圆域中预测结果与有限元法高度一致。
- 适合需要高精度物理模拟但传统方法难处理的场景。
物理信息神经网络为求解复杂物理中的微分方程提供了新路径，但在求解亥姆霍兹方程时受限于梯度消失问题。本文提出一种新方法，将二维亥姆霍兹方程在给定边界条件下的求解转化为无约束优化问题。通过变分插值技术和R函数理论，构建一个训练前即满足边界条件的神经网络试解。该方法首先应用于矩形域，后扩展至圆形和椭圆域。预测声场与二维有限元方法结果对比，三类几何下均表现出良好一致性。同时讨论了方法的局限性及其改进方向。
原文摘要 · Abstract (English)
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.
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