PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh
  Representations
Authors: name: Namgyu Kang, name: Jaemin Oh, name: Youngjoon Hong, name: Eunbyung Park

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 limited
accuracy due to the spectral bias of Multi-Layer Perceptrons (MLPs), which
struggle to effectively learn high-frequency and non-linear components.
Recently, parametric mesh representations in combination with neural networks
have been investigated as a promising approach to eliminate the inductive
biases of neural networks. However, they usually require very high-resolution
grids and a large number of collocation points to achieve high accuracy while
avoiding overfitting issues. In addition, the fixed positions of the mesh
parameters restrict their flexibility, making it challenging to accurately
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. Our project page is available at
https://namgyukang.github.io/Physics-Informed-Gaussians/