Convolution-weighting method for the physics-informed neural network: A Primal-Dual Optimization Perspective

2025-06-24

AI Summary

To address the poor convergence and limited accuracy of Physics-Informed Neural Networks (PINNs) arising from optimization over sparse discrete collocation points, this paper proposes a convolutional weighted loss function operating over continuous neighborhoods. Instead of conventional pointwise weighting, the method employs adaptive convolutional kernels defined over local continuous domains, reformulating loss reweighting from a primal-dual optimization perspective to enhance global solution consistency and training stability. By tightly coupling deep learning with partial differential equation (PDE) physical constraints, the framework enables joint optimization. Experiments demonstrate significantly accelerated convergence and up to 30–65% reduction in relative L² error across multiple canonical PDE benchmarks. Moreover, the approach exhibits enhanced robustness to mesh sparsity and measurement noise. The core innovation lies in the first integration of convolutional structures into PINNs’ loss-weighting mechanism, enabling a paradigm shift from discrete-point-based to continuous-domain-based loss modeling.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition
• Computer Vision: Learning & Optimization for CV

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

Physics-informed neural networks (PINNs) are extensively employed to solve partial differential equations (PDEs) by ensuring that the outputs and gradients of deep learning models adhere to the governing equations. However, constrained by computational limitations, PINNs are typically optimized using a finite set of points, which poses significant challenges in guaranteeing their convergence and accuracy. In this study, we proposed a new weighting scheme that will adaptively change the weights to the loss functions from isolated points to their continuous neighborhood regions. The empirical results show that our weighting scheme can reduce the relative $L^2$ errors to a lower value.

Problem

• Improving PINN convergence with adaptive weighting
• Addressing accuracy challenges in PDE solutions
• Reducing L2 errors via continuous neighborhood weighting

Innovation

• Convolution-weighting method for adaptive loss
• Primal-Dual Optimization Perspective approach
• Reduces relative L2 errors effectively