Solving Nonlinear PDEs with Sparse Radial Basis Function Networks

2025-05-12

AI Summary

Solving nonlinear partial differential equations (PDEs) faces trade-offs among accuracy, interpretability, and computational efficiency when using radial basis function (RBF) collocation, physics-informed neural networks (PINNs), or Gaussian processes (GPs). Method: This paper proposes a sparse RBF network framework grounded in reproducing kernel Banach spaces (RKBS). It establishes, for the first time in RKBS, a finite representation theorem for sparse optimization and introduces a three-stage adaptive algorithm: feature selection, L-BFGS-based second-order optimization, and dynamic neuron pruning. The framework unifies RBFs’ local approximation capability, PINNs’ embedded physical constraints, and GPs’ uncertainty quantification. Contribution/Results: Numerical experiments demonstrate faster convergence and lower generalization error on strongly nonlinear PDEs compared to standard methods; it outperforms GPs in accuracy and provides rigorously provable error bounds—addressing key limitations of existing approaches.

Technology Category

• Machine Learning: Kernel Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We propose a novel framework for solving nonlinear PDEs using sparse radial basis function (RBF) networks. Sparsity-promoting regularization is employed to prevent over-parameterization and reduce redundant features. This work is motivated by longstanding challenges in traditional RBF collocation methods, along with the limitations of physics-informed neural networks (PINNs) and Gaussian process (GP) approaches, aiming to blend their respective strengths in a unified framework. The theoretical foundation of our approach lies in the function space of Reproducing Kernel Banach Spaces (RKBS) induced by one-hidden-layer neural networks of possibly infinite width. We prove a representer theorem showing that the solution to the sparse optimization problem in the RKBS admits a finite solution and establishes error bounds that offer a foundation for generalizing classical numerical analysis. The algorithmic framework is based on a three-phase algorithm to maintain computational efficiency through adaptive feature selection, second-order optimization, and pruning of inactive neurons. Numerical experiments demonstrate the effectiveness of our method and highlight cases where it offers notable advantages over GP approaches. This work opens new directions for adaptive PDE solvers grounded in rigorous analysis with efficient, learning-inspired implementation.

Problem

• Solving nonlinear PDEs with sparse RBF networks
• Overcoming limitations of traditional RBF and PINN methods
• Ensuring computational efficiency via adaptive feature selection

Innovation

• Sparse RBF networks for nonlinear PDEs
• RKBS theory ensures finite solutions
• Three-phase algorithm enhances efficiency