PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations
PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations
物理学计算技术、计算机技术
Youngjoon Hong,Namgyu Kang,Eunbyung Park,Jaemin Oh.PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations[EB/OL].(2024-12-08)[2025-10-26].https://arxiv.org/abs/2412.05994.点此复制
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/
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