PIG：物理資訊高斯適應性參數網格表示法

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

利用神經網絡來近似偏微分方程（PDEs）已經取得了顯著的進展，其中物理信息神經網絡（PINNs）發揮了重要作用。儘管PINNs具有直觀的優化框架和實現各種PDEs的靈活性，但由於多層感知器（MLPs）存在頻譜偏差，難以有效學習高頻和非線性組件，因此PINNs通常存在精度有限的問題。最近，參數網格表示法結合神經網絡被研究作為消除神經網絡歸納偏見的有前途方法。然而，這些方法通常需要非常高分辨率的網格和大量的共點以實現高精度，同時避免過度擬合問題。此外，網格參數的固定位置限制了其靈活性，使得準確近似複雜PDEs具有挑戰性。為了克服這些限制，我們提出了物理信息高斯模型（PIGs），它結合了使用高斯函數的特徵嵌入和輕量級神經網絡。我們的方法使用每個高斯函數的均值和變異數的可訓練參數，允許在訓練期間動態調整它們的位置和形狀。這種適應性使得我們的模型能夠最佳地近似PDE解，與具有固定參數位置的模型不同。此外，所提出的方法保持了PINNs中使用的相同優化框架，使我們能夠受益於它們的優秀特性。實驗結果顯示我們的模型在各種PDEs上具有競爭力的性能，展示了其作為解決複雜PDEs的強大工具的潛力。我們的項目頁面位於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/