Curriculum Learning of Physics-Informed Neural Networks based on Spatial Correlation

2026-05-14

AI Summary

This work addresses the instability and convergence failure of physics-informed neural networks (PINNs) when solving strongly spatially coupled boundary value problems, which stem from high-dimensional non-convex loss landscapes, imbalanced multi-objective optimization, and inefficient information propagation. To overcome these challenges, we propose a spatial correlation–based curriculum learning framework that, for the first time, leverages subdomain coupling to guide boundary information inward via spatial causal weighting. The method enforces cross-regional low-frequency consistency constraints to enhance global coherence and incorporates an adaptive region-wise loss reweighting mechanism to balance local residuals. Evaluated on standard PDE benchmarks, our approach significantly reduces training failure rates and improves both overall solution accuracy and high-frequency detail recovery, all at comparable computational cost.

Abstract

Physics-Informed Neural Networks (PINNs) combine deep learning with physical constraints for solving partial differential equations (PDEs), and are widely applied in fluid mechanics, heat transfer, and solid mechanics. However, PINN training still suffers from high-dimensional non-convex loss landscapes, imbalanced multiobjective constraints, and ineffective information propagation. Existing curriculum learning and causality-guided strategies improve training stability, but mainly focus on temporal or parametric progression, lacking explicit treatment of spatial information propagation and inter-region consistency. Moreover, they are not directly applicable to boundary value problems (BVPs) with strong spatial coupling. To address this issue, we propose a spatially correlated curriculum learning framework for PINNs. To the best of our knowledge, this is the first work to address PINN training difficulties from the perspective of spatial coupling among subregions. First, spatial causal weights guide information from near-boundary regions inward, reducing optimization failures and spurious convergence. Second, a low-frequency information bridge enforces pseudo-label-based consistency across spatially separated regions, suppressing global low-frequency drift. Third, a region-adaptive reweighting strategy adjusts subregion losses to reduce local residuals and recover high-frequency details. Experiments on PDE benchmarks show that, under comparable computational cost, the proposed method alleviates training failures and improves solution accuracy. The code is available at https://github.com/pigofmomo/CurriculumLearningPINN.

Problem

• Physics-Informed Neural Networks
• Curriculum Learning
• Spatial Correlation
• Boundary Value Problems
• Partial Differential Equations

Innovation

• spatial correlation
• curriculum learning
• Physics-Informed Neural Networks
• boundary value problems
• region-adaptive reweighting