# Solving Partial Differential Equations with Random Feature Models

arXiv: https://arxiv.org/abs/2501.00288 · HTML: https://mlsanity.com/paper/2501.00288/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 36 references that are in the map.
- First posted on arXiv 31 Dec 2024.

## Builds on (most cited first, 21 of 21)

- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Just Interpolate: Kernel "Ridgeless" Regression Can Generalize](https://mlsanity.com/paper/1808.00387/) (2018)
- [To understand deep learning we need to understand kernel learning](https://mlsanity.com/paper/1802.01396/) (2018)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [Frequency Bias in Neural Networks for Input of Non-Uniform Density](https://mlsanity.com/paper/2003.04560/) (2020)
- [The Convergence Rate of Neural Networks for Learned Functions of Different Frequencies](https://mlsanity.com/paper/1906.00425/) (2019)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't](https://mlsanity.com/paper/2009.10713/) (2020)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [Random vector functional link network: recent developments, applications, and future directions](https://mlsanity.com/paper/2203.11316/) (2022)
- [But How Does It Work in Theory? Linear SVM with Random Features](https://mlsanity.com/paper/1809.04481/) (2018)
- [Error Bounds for Learning with Vector-Valued Random Features](https://mlsanity.com/paper/2305.17170/) (2023)
- [Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs](https://mlsanity.com/paper/2305.04962/) (2023)
- [An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensions](https://mlsanity.com/paper/2309.07049/) (2023)
- [Random Vector Functional Link Networks for Function Approximation on Manifolds](https://mlsanity.com/paper/2007.15776/) (2020)
- [Solving High Frequency and Multi-Scale PDEs with Gaussian Processes](https://mlsanity.com/paper/2311.04465/) (2023)
- [Toward Efficient Kernel-Based Solvers for Nonlinear PDEs](https://mlsanity.com/paper/2410.11165/) (2024)
- [Differentially Private Random Feature Model](https://mlsanity.com/paper/2412.04785/) (2024)

## Built on by (most cited first, 6 of 6)

- [Sparse RBF Networks for PDEs and nonlocal equations: function space theory, operator calculus, and training algorithms](https://mlsanity.com/paper/2601.17562/) (2026)
- [FastLSQ: Solving PDEs in One Shot via Fourier Features with Exact Analytical Derivatives](https://mlsanity.com/paper/2602.10541/) (2026)
- [MetaColloc: Optimization-Free PDE Solving via Meta-Learned Basis Functions](https://mlsanity.com/paper/2605.12368/) (2026)
- [Frequency Shift Physics-Informed Extreme Learning Machine for Solving High-Frequency Partial Differential Equations](https://mlsanity.com/paper/2607.01694/) (2026)
- [Accelerated Learning of High Dimensional Functions with a Tensor-Featured Training Network](https://mlsanity.com/paper/2608.10351/) (2026)
- [Physics Informed Random Feature Neural Networks for Solving PDEs](https://mlsanity.com/paper/2609.16406/) (2026)

## Abstract (arXiv)

Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and kernel method. In this paper, we introduce a random feature based framework toward efficiently solving PDEs. Random feature method was originally proposed to approximate large-scale kernel machines and can be viewed as a shallow neural network as well. We provide an error analysis for our proposed method along with comprehensive numerical results on several PDE benchmarks. In contrast to the state-of-the-art solvers that face challenges with a large number of collocation points, our proposed method reduces the computational complexity. Moreover, the implementation of our method is simple and does not require additional computational resources. Due to the theoretical guarantee and advantages in computation, our approach is proven to be efficient for solving PDEs.