mlsanityData to 7 Oct 2026

The Random Feature Model for Input-Output Maps between Banach Spaces

arXiv 2005.10224 ↗First posted 20 May 2020math.NA cs.LG cs.NAcited by 90

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Abstract (arXiv)

Well known to the machine learning community, the random feature model is a parametric approximation to kernel interpolation or regression methods. It is typically used to approximate functions mapping a finite-dimensional input space to the real line. In this paper, we instead propose a methodology for use of the random feature model as a data-driven surrogate for operators that map an input Banach space to an output Banach space. Although the methodology is quite general, we consider operators defined by partial differential equations (PDEs); here, the inputs and outputs are themselves functions, with the input parameters being functions required to specify the problem, such as initial data or coefficients, and the outputs being solutions of the problem. Upon discretization, the model inherits several desirable attributes from this infinite-dimensional viewpoint, including mesh-invariant approximation error with respect to the true PDE solution map and the capability to be trained at one mesh resolution and then deployed at different mesh resolutions. We view the random feature model as a non-intrusive data-driven emulator, provide a mathematical framework for its interpretation, and demonstrate its ability to efficiently and accurately approximate the nonlinear parameter-to-solution maps of two prototypical PDEs arising in physical science and engineering applications: viscous Burgers' equation and a variable coefficient elliptic equation.

Key facts. Cited by 90 papers in the map (data through 7 Oct 2026). First posted on arXiv 20 May 2020.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 15; 2022: 19; 2023: 19; 2024: 14; 2025: 14; 2026: 9.

90 papers.

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  1. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  2. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  3. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  4. 2021Choose a Transformer: Fourier or Galerkincited by 333
  5. 2022Transformer for Partial Differential Equations' Operator Learningcited by 270
  6. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
  7. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 128
  8. 2024Operator Learning: Algorithms and Analysiscited by 84
  9. 2022NOMAD: Nonlinear Manifold Decoders for Operator Learningcited by 81
  10. 2023Score-based Diffusion Models in Function Spacecited by 80
  11. 2024On conditional diffusion models for PDE simulationscited by 56
  12. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
38 more
  1. 2022Generative Adversarial Neural Operatorscited by 51
  2. 2022Deep Nonparametric Estimation of Operators between Infinite Dimensional Spacescited by 45
  3. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
  4. 2022Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learningcited by 43
  5. 2022Mitigating spectral bias for the multiscale operator learningcited by 43
  6. 2023Nonlocality and Nonlinearity Implies Universality in Operator Learningcited by 43
  7. 2021A Framework for Machine Learning of Model Error in Dynamical Systemscited by 37
  8. 2023Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processescited by 31
  9. 2023Koopman neural operator as a mesh-free solver of non-linear partial differential equationscited by 29
  10. 2021Two-layer neural networks with values in a Banach spacecited by 26
  11. 2021Learning the optimal Tikhonov regularizer for inverse problemscited by 26
  12. 2022Partial Differential Equations Meet Deep Neural Networks: A Surveycited by 25
  13. 2022Incremental Spatial and Spectral Learning of Neural Operators for Solving Large-Scale PDEscited by 25
  14. 2022Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEscited by 24
  15. 2022Transformer Meets Boundary Value Inverse Problemscited by 22
  16. 2023A Neural PDE Solver with Temporal Stencil Modelingcited by 22
  17. 2021Reduced operator inference for nonlinear partial differential equationscited by 21
  18. 2024Data Complexity Estimates for Operator Learningcited by 21
  19. 2023Sampling weights of deep neural networkscited by 20
  20. 2023Deep Operator Learning Lessens the Curse of Dimensionality for PDEscited by 18
  21. 2023Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operatorscited by 18
  22. 2023RecFNO: a resolution-invariant flow and heat field reconstruction method from sparse observations via Fourier neural operatorcited by 17
  23. 2021Learning High-Dimensional Parametric Maps via Reduced Basis Adaptive Residual Networkscited by 16
  24. 2024RandONet: Shallow-Networks with Random Projections for learning linear and nonlinear operatorscited by 16
  25. 2023Energy-Dissipative Evolutionary Deep Operator Neural Networkscited by 14
  26. 2023TransNet: Transferable Neural Networks for Partial Differential Equationscited by 13
  27. 2023Learning About Structural Errors in Models of Complex Dynamical Systemscited by 13
  28. 2023Learning Only On Boundaries: a Physics-Informed Neural operator for Solving Parametric Partial Differential Equations in Complex Geometriescited by 12
  29. 2023Universal approximation property of Banach space-valued random feature models including random neural networkscited by 12
  30. 2022Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEscited by 11
  31. 2022Catch-22s of reservoir computingcited by 11
  32. 2024Universal Functional Regression with Neural Operator Flowscited by 9
  33. 2022Local approximation of operatorscited by 8
  34. 2024Hyperparameter Optimization for Randomized Algorithms: A Case Study on Random Featurescited by 7
  35. 2025Learning dynamical systems with hit-and-run random feature mapscited by 7
  36. 2021Statistical Depth Meets Machine Learning: Kernel Mean Embeddings and Depth in Functional Data Analysiscited by 6
  37. 2021A generalization of the randomized singular value decompositioncited by 6
  38. 2022Semi-supervised Learning of Partial Differential Operators and Dynamical Flowscited by 6

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