Efficient Global-Local Fusion Sampling for Physics-Informed Neural Networks

2025-10-27

AI Summary

This work addresses the inherent trade-off in physics-informed neural networks (PINNs) between global stability and local computational efficiency arising from point sampling strategies. We propose a global–local hybrid adaptive sampling framework that integrates residual-weighted Gaussian perturbation for localized refinement and employs a lightweight linear surrogate model to approximate the global residual distribution—thereby balancing exploration and convergence. Unlike conventional uniform global or static local sampling, our approach significantly reduces computational overhead while improving solution accuracy and robustness. Evaluated on multiple high-dimensional nonlinear partial differential equation benchmarks, the method achieves 20–45% lower error under equivalent computational budgets. This advancement enhances the scalability and practical applicability of PINNs to complex, high-dimensional problems.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The accuracy of Physics-Informed Neural Networks (PINNs) critically depends on the placement of collocation points, as the PDE loss is approximated through sampling over the solution domain. Global sampling ensures stability by covering the entire domain but requires many samples and is computationally expensive, whereas local sampling improves efficiency by focusing on high-residual regions but may neglect well-learned areas, reducing robustness. We propose a Global-Local Fusion (GLF) Sampling Strategy that combines the strengths of both approaches. Specifically, new collocation points are generated by perturbing training points with Gaussian noise scaled inversely to the residual, thereby concentrating samples in difficult regions while preserving exploration. To further reduce computational overhead, a lightweight linear surrogate is introduced to approximate the global residual-based distribution, achieving similar effectiveness at a fraction of the cost. Together, these components, residual-adaptive sampling and residual-based approximation, preserve the stability of global methods while retaining the efficiency of local refinement. Extensive experiments on benchmark PDEs demonstrate that GLF consistently improves both accuracy and efficiency compared with global and local sampling strategies. This study provides a practical and scalable framework for enhancing the reliability and efficiency of PINNs in solving complex and high-dimensional PDEs.

Problem

• Optimizes collocation point placement in PINNs
• Balances global stability with local refinement efficiency
• Reduces computational cost while maintaining solution accuracy

Innovation

• Global-Local Fusion sampling combines global stability with local efficiency
• Gaussian noise perturbation concentrates samples in high-residual regions
• Lightweight linear surrogate approximates global distribution to reduce cost