mlsanityData to 7 Oct 2026

Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality

arXiv 2106.08900 ↗First posted 14 Jun 2021cs.LG math.PR q-fin.MFcited by 34

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Abstract (arXiv)

This article investigates the use of random feature neural networks for learning Kolmogorov partial (integro-)differential equations associated to Black-Scholes and more general exponential L'evy models. Random feature neural networks are single-hidden-layer feedforward neural networks in which only the output weights are trainable. This makes training particularly simple, but (a priori) reduces expressivity. Interestingly, this is not the case for Black-Scholes type PDEs, as we show here. We derive bounds for the prediction error of random neural networks for learning sufficiently non-degenerate Black-Scholes type models. A full error analysis is provided and it is shown that the derived bounds do not suffer from the curse of dimensionality. We also investigate an application of these results to basket options and validate the bounds numerically. These results prove that neural networks are able to textitlearn solutions to Black-Scholes type PDEs without the curse of dimensionality. In addition, this provides an example of a relevant learning problem in which random feature neural networks are provably efficient.

Key facts. Cited by 34 papers in the map (data through 7 Oct 2026). Builds on 28 of its 67 references that are in the map. First posted on arXiv 14 Jun 2021.

Builds on

28 of 67 references.

References · 28

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 3; 2023: 9; 2024: 9; 2025: 7; 2026: 5.

34 papers.

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  1. 2023An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensionscited by 21
  2. 2022Chefs' Random Tables: Non-Trigonometric Random Featurescited by 20
  3. 2021On the rate of convergence of a classifier based on a Transformer encodercited by 16
  4. 2023Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirscited by 15
  5. 2023Universal approximation property of Banach space-valued random feature models including random neural networkscited by 12
  6. 2024Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEscited by 12
  7. 2022Chaotic Hedging with Iterated Integrals and Neural Networkscited by 10
  8. 2022Deep neural network expressivity for optimal stopping problemscited by 9
  9. 2023Infinite-dimensional reservoir computingcited by 7
  10. 2024An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learningcited by 6
  11. 2024Solving stochastic partial differential equations using neural networks in the Wiener chaos expansioncited by 6
  12. 2023Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spacescited by 5
22 more
  1. 2025A brief review of the Deep BSDE method for solving high-dimensional partial differential equationscited by 5
  2. 2025Feedback-driven recurrent quantum neural network universalitycited by 5
  3. 2023FAVOR#: Sharp Attention Kernel Approximations via New Classes of Positive Random Featurescited by 4
  4. 2024Universal randomised signatures for generative time series modellingcited by 4
  5. 2025Random Feature Representation Boostingcited by 4
  6. 2024Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in Lp-sensecited by 3
  7. 2024RandNet-Parareal: a time-parallel PDE solver using Random Neural Networkscited by 3
  8. 2023Regret-Optimal Federated Transfer Learning for Kernel Regression with Applications in American Option Pricingcited by 2
  9. 2023On the rate of convergence of an over-parametrized Transformer classifier learned by gradient descentcited by 2
  10. 2023Unsupervised Random Quantum Networks for PDEscited by 1
  11. 2024Deep Learning without Global Optimization by Random Fourier Neural Networkscited by 1
  12. 2025Beyond Universal Approximation Theorems: Algorithmic Uniform Approximation by Neural Networks Trained with Noisy Datacited by 1
  13. 2024Linear Stability Analysis of Physics-Informed Random Projection Neural Networks for ODEscited by 0
  14. 2024Kernel Approximation using Analog In-Memory Computingcited by 0
  15. 2025Error analysis for the deep Kolmogorov methodcited by 0
  16. 2025A Neural Network Algorithm for KL Divergence Estimation with Quantitative Error Boundscited by 0
  17. 2025Supervised Learning of Random Neural Architectures Structured by Latent Random Fields on Compact Boundaryless Multiply-Connected Manifoldscited by 0
  18. 2026Lipschitz bounds for integral kernelscited by 0
  19. 2026Random Neural Network Expressivity for Non-Linear Partial Differential Equationscited by 0
  20. 2026NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processescited by 0
  21. 2026Spectral Convergence of Random Feature Method in Multiple Dimensionscited by 0
  22. 2026Residual neural networks overcome the curse of dimensionality for semilinear heat equationscited by 0