CGMPINN: Curriculum-Guided Physics-Informed Neural Network Based on Gaussian Mixture Model

The team from Xi'an Jiaotong University proposed CGMPINN (Curriculum-Guided Gaussian Mixture Physics-Informed Neural Network), which models the PDE residual distribution using a Gaussian Mixture Model to implement a spatially adaptive curriculum learning strategy, reducing the relative L2 error by up to 97.8% on six benchmark PDE problems.

Physics-Informed Neural Networks (PINNs) are mesh-free PDE solving frameworks, but traditional PINNs face issues such as ill-conditioned gradients, spectral bias, and convergence difficulties, especially performing poorly in problems with strong nonlinearity, steep gradients, or multi-scales. The root cause lies in the uneven distribution of the PDE solution space; a one-size-fits-all training approach easily leads to local optima or neglect of difficult regions.

CGMPINN draws on the concept of curriculum learning, allowing the model to gradually transition from simple regions to complex ones. By modeling the PDE residual distribution using a Gaussian Mixture Model (GMM), it identifies clusters of regions with different difficulty levels—regions with large residuals are more difficult to learn, thus implementing a spatially adaptive curriculum strategy.

CGMPINN includes two key mechanisms:

1. Spatial Difficulty-Aware Curriculum: Periodically fits GMM to divide difficulty regions, focusing on simple regions in the early training stage and gradually shifting to difficult regions;
2. Accuracy-Guided Variance Modulation: Reduces the loss weight of clusters with large variance (unstable prediction regions) to minimize noise interference. Both share curriculum parameters and can be combined with adaptive loss balancing methods.

The research team proved the key properties of CGMPINN:

• Sublinear Convergence: The gradient norm of the curriculum-weighted loss converges sublinearly, ensuring optimization stability;
• Loss Equivalence: The curriculum-weighted PDE loss is uniformly equivalent to the standard loss under certain conditions;
• Generalization Bounds: Derived an upper bound of generalization error including weight bias, revealing the impact of curriculum learning on generalization.

The team validated the performance on six types of PDE problems, including elliptic (Poisson equation), parabolic (Heat equation), hyperbolic (Wave equation), convection-dominated (Advection-Diffusion), nonlinear reaction-diffusion (Fisher-KPP equation), etc. The results show that CGMPINN achieved the lowest relative L2 error in all cases, reducing the error by up to 97.8% compared to standard PINNs, with comparable computational cost.

The project provides a complete PyTorch implementation, including:

• Core CGMPINN methods;
• Baseline methods (standard PINN, gPINN, lbPINN, etc.);
• Ablation experiment variants. Each benchmark problem has an independent code and parameter directory, and pre-trained models can directly reproduce the results.

CGMPINN provides a solution to PINN training challenges, enhancing its applicability in complex physical problems. It is of great significance to the fields of computational physics, engineering simulation, and scientific machine learning, and the open-source implementation provides a benchmark tool for the community. This method integrates classical machine learning techniques with physical insights, advancing the theoretical frontier of PINNs and providing a powerful tool for solving practical complex problems.