PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations

Namgyu Kang, Jaemin Oh, Youngjoon Hong, Eunbyung Park

Keywords: Gaussians, Physics-informed Deep Learning

Abstract: The 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 low accuracy due to the spectral bias of Multi-Layer Perceptrons (MLPs), which struggle to effectively learn high-frequency and non-linear components. Recently, the parametric mesh representations have been investigated as a promising approach to effectively eliminating the inductive biases of neural networks. However, they often require very high-resolution grids and a large number of collocation points to achieve high accuracy while avoiding overfitting issues. In addition, these are limited by the fixed positions of the mesh parameters, hindering their ability to approximate complex PDEs. 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.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

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Submission Number: 5707