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

arXiv: https://arxiv.org/abs/2106.08900 · HTML: https://mlsanity.com/paper/2106.08900/

## Key facts

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

## Builds on (most cited first, 28 of 28)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [The generalization error of random features regression: Precise asymptotics and double descent curve](https://mlsanity.com/paper/1908.05355/) (2019)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [Generalization error of random features and kernel methods: hypercontractivity and kernel matrix concentration](https://mlsanity.com/paper/2101.10588/) (2021)
- [Deep Hedging](https://mlsanity.com/paper/1802.03042/) (2018)
- [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)
- [Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with  ℓ1  and  ℓ0  Controls](https://mlsanity.com/paper/1607.07819/) (2016)
- [A Priori Estimates of the Population Risk for Two-layer Neural Networks](https://mlsanity.com/paper/1810.06397/) (2018)
- [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019)
- [An overview on deep learning-based approximation methods for partial differential equations](https://mlsanity.com/paper/2012.12348/) (2020)
- [The Modern Mathematics of Deep Learning](https://mlsanity.com/paper/2105.04026/) (2021)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Approximation Bounds for Random Neural Networks and Reservoir Systems](https://mlsanity.com/paper/2002.05933/) (2020)
- [Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations](https://mlsanity.com/paper/1809.03062/) (2018)
- [Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory](https://mlsanity.com/paper/2006.15733/) (2020)
- [Neural networks for option pricing and hedging: a literature review](https://mlsanity.com/paper/1911.05620/) (2019)
- [On the Banach spaces associated with multi-layer ReLU networks: Function representation, approximation theory and gradient descent dynamics](https://mlsanity.com/paper/2007.15623/) (2020)
- [Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games: II -- The Finite Horizon Case](https://mlsanity.com/paper/1908.01613/) (2019)
- [Learning with SGD and Random Features](https://mlsanity.com/paper/1807.06343/) (2018)
- [A generative adversarial network approach to calibration of local stochastic volatility models](https://mlsanity.com/paper/2005.02505/) (2020)
- [Neural network approximation and estimation of classifiers with classification boundary in a Barron class](https://mlsanity.com/paper/2011.09363/) (2020)
- [Neural networks-based algorithms for stochastic control and PDEs in finance](https://mlsanity.com/paper/2101.08068/) (2021)
- [Approximation and Estimation for High-Dimensional Deep Learning Networks](https://mlsanity.com/paper/1809.03090/) (2018)
- [Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems](https://mlsanity.com/paper/1903.06652/) (2019)
- [Uniform error estimates for artificial neural network approximations for heat equations](https://mlsanity.com/paper/1911.09647/) (2019)

## Built on by (most cited first, 30 of 34)

- [Chefs' Random Tables: Non-Trigonometric Random Features](https://mlsanity.com/paper/2205.15317/) (2022)
- [An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensions](https://mlsanity.com/paper/2309.07049/) (2023)
- [On the rate of convergence of a classifier based on a Transformer encoder](https://mlsanity.com/paper/2111.14574/) (2021)
- [Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirs](https://mlsanity.com/paper/2307.12904/) (2023)
- [Universal approximation property of Banach space-valued random feature models including random neural networks](https://mlsanity.com/paper/2312.08410/) (2023)
- [Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEs](https://mlsanity.com/paper/2405.05192/) (2024)
- [Chaotic Hedging with Iterated Integrals and Neural Networks](https://mlsanity.com/paper/2209.10166/) (2022)
- [Deep neural network expressivity for optimal stopping problems](https://mlsanity.com/paper/2210.10443/) (2022)
- [Infinite-dimensional reservoir computing](https://mlsanity.com/paper/2304.00490/) (2023)
- [An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learning](https://mlsanity.com/paper/2408.13222/) (2024)
- [Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion](https://mlsanity.com/paper/2411.03384/) (2024)
- [Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spaces](https://mlsanity.com/paper/2312.08461/) (2023)
- [A brief review of the Deep BSDE method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2505.17032/) (2025)
- [Feedback-driven recurrent quantum neural network universality](https://mlsanity.com/paper/2506.16332/) (2025)
- [FAVOR#: Sharp Attention Kernel Approximations via New Classes of Positive Random Features](https://mlsanity.com/paper/2302.00787/) (2023)
- [Universal randomised signatures for generative time series modelling](https://mlsanity.com/paper/2406.10214/) (2024)
- [Random Feature Representation Boosting](https://mlsanity.com/paper/2501.18283/) (2025)
- [Multilevel 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-sense](https://mlsanity.com/paper/2409.20431/) (2024)
- [RandNet-Parareal: a time-parallel PDE solver using Random Neural Networks](https://mlsanity.com/paper/2411.06225/) (2024)
- [Regret-Optimal Federated Transfer Learning for Kernel Regression with Applications in American Option Pricing](https://mlsanity.com/paper/2309.04557/) (2023)
- [On the rate of convergence of an over-parametrized Transformer classifier learned by gradient descent](https://mlsanity.com/paper/2312.17007/) (2023)
- [Beyond Universal Approximation Theorems: Algorithmic Uniform Approximation by Neural Networks Trained with Noisy Data](https://mlsanity.com/paper/2509.00924/) (2025)
- [Unsupervised Random Quantum Networks for PDEs](https://mlsanity.com/paper/2312.14975/) (2023)
- [Deep Learning without Global Optimization by Random Fourier Neural Networks](https://mlsanity.com/paper/2407.11894/) (2024)
- [Linear Stability Analysis of Physics-Informed Random Projection Neural Networks for ODEs](https://mlsanity.com/paper/2408.15393/) (2024)
- [Kernel Approximation using Analog In-Memory Computing](https://mlsanity.com/paper/2411.03375/) (2024)
- [Error analysis for the deep Kolmogorov method](https://mlsanity.com/paper/2508.17167/) (2025)
- [A Neural Network Algorithm for KL Divergence Estimation with Quantitative Error Bounds](https://mlsanity.com/paper/2510.05386/) (2025)
- [Supervised Learning of Random Neural Architectures Structured by Latent Random Fields on Compact Boundaryless Multiply-Connected Manifolds](https://mlsanity.com/paper/2512.10407/) (2025)
- [Lipschitz bounds for integral kernels](https://mlsanity.com/paper/2604.02887/) (2026)

## 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.