PIG：物理信息高斯函数作为自适应参数化网格表示形式

PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations

December 8, 2024

作者: Namgyu Kang, Jaemin Oh, Youngjoon Hong, Eunbyung Park

cs.AI

摘要

利用神经网络逼近偏微分方程（PDE）的方法已经在物理信息神经网络（PINNs）中取得了显著进展。尽管PINNs具有直观的优化框架和实现各种PDE的灵活性，但由于多层感知器（MLPs）的谱偏差，它们往往精度有限，难以有效学习高频和非线性成分。最近，参数化网格表示结合神经网络被研究作为消除神经网络归纳偏见的一种有前途的方法。然而，它们通常需要非常高分辨率的网格和大量的共轭点才能实现高精度，同时避免过拟合问题。此外，网格参数的固定位置限制了它们的灵活性，使得准确逼近复杂PDE变得具有挑战性。为了克服这些限制，我们提出了物理信息高斯模型（PIGs），它使用高斯函数结合轻量级神经网络来组合特征嵌入。我们的方法使用可训练参数来调整每个高斯分布的均值和方差，允许在训练过程中动态调整它们的位置和形状。这种适应性使我们的模型能够最佳地逼近PDE解，不同于具有固定参数位置的模型。此外，所提出的方法保持了PINNs中使用的相同优化框架，使我们能够从它们的优秀特性中受益。实验结果表明，我们的模型在各种PDE上表现出竞争力，展示了其作为解决复杂PDE的强大工具的潜力。我们的项目页面位于https://namgyukang.github.io/Physics-Informed-Gaussians/。

English

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/