# 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

arXiv: https://arxiv.org/abs/1809.03062 · HTML: https://mlsanity.com/paper/1809.03062/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Sep 2018.

## Built on by (most cited first, 30 of 45)

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- [An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensions](https://mlsanity.com/paper/2309.07049/) (2023)
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- [A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations](https://mlsanity.com/paper/2101.01708/) (2021)
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- [Optimal learning of high-dimensional classification problems using deep neural networks](https://mlsanity.com/paper/2112.12555/) (2021)
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- [Stable Recovery of Entangled Weights: Towards Robust Identification of Deep Neural Networks from Minimal Samples](https://mlsanity.com/paper/2101.07150/) (2021)
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- [Quantitative approximation results for complex-valued neural networks](https://mlsanity.com/paper/2102.13092/) (2021)
- [Deep neural network approximation for high-dimensional parabolic Hamilton-Jacobi-Bellman equations](https://mlsanity.com/paper/2103.05744/) (2021)
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- [Stable Learning Using Spiking Neural Networks Equipped With Affine Encoders and Decoders](https://mlsanity.com/paper/2404.04549/) (2024)
- [A brief review of the Deep BSDE method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2505.17032/) (2025)
- [Deep neural network surrogates for non-smooth quantities of interest in shape uncertainty quantification](https://mlsanity.com/paper/2101.07023/) (2021)
- [Metric Entropy Limits on Recurrent Neural Network Learning of Linear Dynamical Systems](https://mlsanity.com/paper/2105.02556/) (2021)
- [A Rescaling-Invariant Lipschitz Bound Based on Path-Metrics for Modern ReLU Network Parameterizations](https://mlsanity.com/paper/2405.15006/) (2024)

## Abstract (arXiv)

The development of new classification and regression algorithms based on empirical risk minimization (ERM) over deep neural network hypothesis classes, coined deep learning, revolutionized the area of artificial intelligence, machine learning, and data analysis. In particular, these methods have been applied to the numerical solution of high-dimensional partial differential equations with great success. Recent simulations indicate that deep learning-based algorithms are capable of overcoming the curse of dimensionality for the numerical solution of Kolmogorov equations, which are widely used in models from engineering, finance, and the natural sciences. The present paper considers under which conditions ERM over a deep neural network hypothesis class approximates the solution of a d-dimensional Kolmogorov equation with affine drift and diffusion coefficients and typical initial values arising from problems in computational finance up to error varepsilon. We establish that, with high probability over draws of training samples, such an approximation can be achieved with both the size of the hypothesis class and the number of training samples scaling only polynomially in d and varepsilon-1. It can be concluded that ERM over deep neural network hypothesis classes overcomes the curse of dimensionality for the numerical solution of linear Kolmogorov equations with affine coefficients.