mlsanityData to 14 Oct 2026

Solving Partial Differential Equations with Random Feature Models

arXiv 2501.00288 ↗First posted 31 Dec 2024math.NA cs.LG cs.NAcited by 6

Not on the map. Why?

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.

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

21 of 36 references.

References · 21

  1. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,200
  2. 2018On the Spectral Bias of Neural Networkscited by 1,341
  3. 2019The generalization error of random features regression: Precise asymptotics and double descent curvecited by 413
  4. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 284
  5. 2018Just Interpolate: Kernel "Ridgeless" Regression Can Generalizecited by 190
  6. 2018To understand deep learning we need to understand kernel learningcited by 187
  7. 2016Generalization Properties of Learning with Random Featurescited by 164
  8. 2020Frequency Bias in Neural Networks for Input of Non-Uniform Densitycited by 140
  9. 2019The Convergence Rate of Neural Networks for Learned Functions of Different Frequenciescited by 135
  10. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 129
  11. 2020Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don'tcited by 95
  12. 2020Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equationscited by 81
9 more
  1. 2022Random vector functional link network: recent developments, applications, and future directionscited by 33
  2. 2018But How Does It Work in Theory? Linear SVM with Random Featurescited by 28
  3. 2023Error Bounds for Learning with Vector-Valued Random Featurescited by 23
  4. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  5. 2023An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensionscited by 21
  6. 2020Random Vector Functional Link Networks for Function Approximation on Manifoldscited by 19
  7. 2023Solving High Frequency and Multi-Scale PDEs with Gaussian Processescited by 14
  8. 2024Toward Efficient Kernel-Based Solvers for Nonlinear PDEscited by 7
  9. 2024Differentially Private Random Feature Modelcited by 2

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 6.

6 papers.

  1. 2026Sparse RBF Networks for PDEs and nonlocal equations: function space theory, operator calculus, and training algorithmscited by 1
  2. 2026FastLSQ: Solving PDEs in One Shot via Fourier Features with Exact Analytical Derivativescited by 0
  3. 2026MetaColloc: Optimization-Free PDE Solving via Meta-Learned Basis Functionscited by 0
  4. 2026Frequency Shift Physics-Informed Extreme Learning Machine for Solving High-Frequency Partial Differential Equationscited by 0
  5. 2026Accelerated Learning of High Dimensional Functions with a Tensor-Featured Training Networkcited by 0
  6. 2026Physics Informed Random Feature Neural Networks for Solving PDEscited by 0