PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations

TL;DR

Physics-Informed Gaussians (PIG) address PINN limitations by replacing fixed parametric grids with trainable Gaussian bases whose means $\mu_i$ and covariances $\Sigma_i$ are updated during training to adaptively allocate representational capacity. A lightweight MLP refines Gaussian embeddings to produce PDE solutions, preserving the PINN optimization framework while enabling rapid convergence and parameter-efficient accuracy. The approach is supported by a universal approximation result for PIGs and demonstrated across multiple challenging PDEs (e.g., Allen-Cahn, Helmholtz, Klein-Gordon, nonlinear diffusion, and flow mixing), where PIGs achieve competitive or superior accuracy with substantially fewer parameters than baselines. This adaptive, mesh-like representation holds promise for scalable, high-dimensional PDE solving with improved efficiency and robustness in practice.

Abstract

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/

Figures (19)

• Figure 1: Training visualization of the Allen-Cahn equation (400, 800, 1200, 4000 training iterations): Each Gaussian is displayed as the ellipsoids, exhibiting different positions and shapes according to the Gaussian parameters, mean and covariance. Since we adopt a causal loss wang2024respecting, the solution is gradually approximated from $t=0$ to $t=1$. Note that the Gaussians are densely aligned in the locations where the solution changes abruptly.
• Figure 2: (a) PINN directly takes input coordinates (four collocation points) as inputs and produces outputs. (b) Parametric grids first map input coordinates to output feature vectors. Each vertex in the grids holds learnable parameters, and output features are extracted through interpolation schemes. (c) The proposed PIG consists of numerous Gaussians moving around within the input domain, and their shapes change dynamically during training. Each Gaussian has learnable parameters, and a feature vector for an input coordinate is the weighted sum of the learnable parameters based on the distance to the Gaussians.
• Figure 3: PIG as a neural network.
• Figure 4: Allen-Cahn Equation. Reference solution and absolute error maps of PIG and one of the state-of-the-art methods (JAX-PI) to Allen-Cahn Equation (x-axis: $t$, y-axis: $x$). The rightmost depicts a relative $L^2$ error curve during the training process (x-axis: iterations, y-axis: $L^2$ error). The experiment was conducted with three different seeds, and the best relative $L^2$ error of PIG is $5.93 \times 10^{-5}$.
• Figure 5: 2D Helmholtz Equation. Reference solution and absolute error maps of PIG and one of the state-of-the-art methods (PIXEL) to 2D Helmholtz Equation. The rightmost depicts a relative $L^2$ error curve during the training process and the best relative $L^2$ error of PIG is $2.22 \times 10^{-5}$.

Theorems & Definitions (2)

• Theorem 1: park1991universal
• Corollary 1