From Simple to Complex: Curriculum-Guided Physics-Informed Neural Networks via Gaussian Mixture Models

TL;DR

This paper introduces CGMPINN, a curriculum-guided physics-informed neural network that uses Gaussian mixture models to adaptively focus training on easier or harder PDE regions, improving convergence and accuracy.

Contribution

It proposes a novel curriculum learning approach integrated with GMMs for PINNs, providing theoretical guarantees and superior performance on benchmark PDEs.

Findings

CGMPINN achieves up to 97.8% error reduction compared to standard PINNs.

The method demonstrates consistent accuracy improvements across six diverse PDE benchmarks.

Theoretical analysis guarantees convergence and bounds for the curriculum-guided training process.

Abstract

Physics-informed neural networks (PINNs) offer a mesh-free framework for solving partial differential equations (PDEs), yet training often suffers from gradient pathologies, spectral bias, and poor convergence, especially for problems with strong nonlinearity, sharp gradients, or multiscale features. We propose the Curriculum-Guided Gaussian Mixture Physics-Informed Neural Network (CGMPINN), which integrates Gaussian mixture modeling with dynamic curriculum learning. Specifically, a GMM is periodically fitted to the PDE residual distribution to quantify spatially varying learning difficulty. A smooth curriculum schedule progressively shifts training focus from easy to harder regions, while precision-based variance modulation suppresses unreliable clusters during early optimization. This dual curriculum is governed by a shared curriculum parameter and can be combined with self-adaptive loss balancing. We further establish theoretical guarantees, including sublinear convergence of the gradient norm for the induced time-varying loss, uniform equivalence between the curriculum-weighted and standard PDE losses, and a generalization bound with an explicit weighting-induced bias characterization. Experiments on six benchmark PDEs spanning elliptic, parabolic, hyperbolic, advection-dominated, and nonlinear reaction-diffusion types show that CGMPINN consistently achieves the lowest relative $L_2$ and maximum absolute errors among all compared methods, reducing relative $L_2$ error by up to 97.8\% over the standard PINN at comparable cost. Our code is publicly available at https://github.com/Mathematics-Yang/CGMPINN.