# Solving Nonlinear PDEs with Sparse Radial Basis Function Networks

arXiv: https://arxiv.org/abs/2505.07765 · HTML: https://mlsanity.com/paper/2505.07765/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 63 references that are in the map.
- First posted on arXiv 12 May 2025.

## Builds on (most cited first, 23 of 23)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Coordinate Descent Algorithms](https://mlsanity.com/paper/1502.04759/) (2015)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Understanding neural networks with reproducing kernel Banach spaces](https://mlsanity.com/paper/2109.09710/) (2021)
- [Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory](https://mlsanity.com/paper/2006.15733/) (2020)
- [Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks](https://mlsanity.com/paper/2109.08844/) (2021)
- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructions](https://mlsanity.com/paper/1901.01002/) (2019)
- [Sparsity of solutions for variational inverse problems with finite-dimensional data](https://mlsanity.com/paper/1809.05045/) (2018)
- [On Representer Theorems and Convex Regularization](https://mlsanity.com/paper/1806.09810/) (2018)
- [Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs](https://mlsanity.com/paper/2305.04962/) (2023)
- [TransNet: Transferable Neural Networks for Partial Differential Equations](https://mlsanity.com/paper/2301.11701/) (2023)
- [Numerical methods for nonlocal and fractional models](https://mlsanity.com/paper/2002.01401/) (2020)
- [Stochastic Gauss-Newton Algorithms for Nonconvex Compositional Optimization](https://mlsanity.com/paper/2002.07290/) (2020)
- [Nonconvex regularization for sparse neural networks](https://mlsanity.com/paper/2004.11515/) (2020)
- [On the relationship between multivariate splines and infinitely-wide neural networks](https://mlsanity.com/paper/2302.03459/) (2023)
- [Function-Space Optimality of Neural Architectures with Multivariate Nonlinearities](https://mlsanity.com/paper/2310.03696/) (2023)
- [Nonuniform random feature models using derivative information](https://mlsanity.com/paper/2410.02132/) (2024)
- [Mirror Descent on Reproducing Kernel Banach Spaces](https://mlsanity.com/paper/2411.11242/) (2024)
- [No-gap second-order conditions for minimization problems in spaces of measures](https://mlsanity.com/paper/2403.12001/) (2024)

## Built on by (most cited first, 3 of 3)

- [AB-PINNs: Adaptive-Basis Physics-Informed Neural Networks for Residual-Driven Domain Decomposition](https://mlsanity.com/paper/2510.08924/) (2025)
- [Bilevel optimization for learning hyperparameters: Application to solving PDEs and inverse problems with Gaussian processes](https://mlsanity.com/paper/2510.05568/) (2025)
- [Sparse RBF Networks for PDEs and nonlocal equations: function space theory, operator calculus, and training algorithms](https://mlsanity.com/paper/2601.17562/) (2026)

## Abstract (arXiv)

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 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.