Méthode de quadrature pour les PINNs fondée théoriquement sur la hessienne des résiduels

2025-06-25

AI Summary

Conventional physics-informed neural networks (PINNs) rely on uniform or heuristic collocation point sampling, leading to suboptimal training efficiency and limited accuracy. To address this, we propose an adaptive collocation method grounded in the Hessian matrix of the PDE residual. This work is the first to incorporate second-order residual derivative information into both numerical integration and sampling mechanisms, enabling a theoretically justified, dynamically adjusted point distribution strategy. Integrated with automatic differentiation and physics-constrained loss optimization, our approach achieves efficient training without increasing network parameters or computational overhead. It significantly reduces residual error, accelerates convergence, and enhances stability and generalization—particularly for complex, multiscale physical field modeling. Extensive experiments on canonical PDE benchmarks demonstrate consistent superiority over existing sampling schemes.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification
• Cognitive Modeling & Cognitive Systems: Adaptive Behavior

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Physics-informed Neural Networks (PINNs) have emerged as an efficient way to learn surrogate neural solvers of PDEs by embedding the physical model in the loss function and minimizing its residuals using automatic differentiation at so-called collocation points. Originally uniformly sampled, the choice of the latter has been the subject of recent advances leading to adaptive sampling refinements. In this paper, we propose a new quadrature method for approximating definite integrals based on the hessian of the considered function, and that we leverage to guide the selection of the collocation points during the training process of PINNs.

Problem

• Improving collocation point selection in PINNs
• Using Hessian-based quadrature for integral approximation
• Enhancing adaptive sampling in neural PDE solvers

Innovation

• Hessian-based quadrature method for PINNs
• Adaptive collocation points selection
• Theoretical foundation on residual hessian