PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations

TL;DR

This paper introduces Physics-Informed Gaussians (PIGs), a novel adaptive parametric mesh method that enhances PDE approximation by dynamically adjusting Gaussian features within a PINNs framework, improving accuracy and flexibility.

Contribution

The paper proposes PIGs, combining Gaussian feature embeddings with neural networks, allowing trainable means and variances for better PDE solution approximation, overcoming fixed mesh limitations.

Findings

Competitive performance across various PDEs

Enhanced adaptability over fixed mesh methods

Maintains PINNs optimization advantages

Abstract

The numerical approximation of partial differential equations (PDEs) using neural networks has seen significant advancements through Physics-Informed Neural Networks (PINNs). Despite their straightforward optimization framework and flexibility in implementing various PDEs, PINNs often suffer from limited accuracy due to the spectral bias of Multi-Layer Perceptrons (MLPs), which struggle to effectively learn high-frequency and nonlinear components. Recently, parametric mesh representations in combination with neural networks have been investigated as a promising approach to eliminate the inductive bias of MLPs. However, they usually require high-resolution grids and a large number of collocation points to achieve high accuracy while avoiding overfitting. In addition, the fixed positions of the mesh parameters restrict their flexibility, making accurate approximation of complex PDEs challenging. To overcome these limitations, we propose Physics-Informed Gaussians (PIGs), which combine feature embeddings using Gaussian functions with a lightweight neural network. Our approach uses trainable parameters for the mean and variance of each Gaussian, allowing for dynamic adjustment of their positions and shapes during training. This adaptability enables our model to optimally approximate PDE solutions, unlike models with fixed parameter positions. Furthermore, the proposed approach maintains the same optimization framework used in PINNs, allowing us to benefit from their excellent properties. Experimental results show the competitive performance of our model across various PDEs, demonstrating its potential as a robust tool for solving complex PDEs. Our project page is available at https://namgyukang.github.io/Physics-Informed-Gaussians/