# A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations

## TL;DR

This paper rigorously proves that deep neural networks can efficiently approximate solutions to high-dimensional semilinear heat equations, overcoming the curse of dimensionality with polynomial growth in parameters.

## Contribution

It provides the first mathematical proof that deep neural networks overcome the curse of dimensionality for a class of nonlinear PDEs, specifically semilinear heat equations.

## Key findings

- Neural network parameters grow polynomially with dimension and accuracy
- Proof relies on multilevel Picard approximations
- Overcomes the curse of dimensionality in nonlinear PDE approximation

## Abstract

Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of high-dimensional nonlinear parabolic partial differential equations (PDEs), which are widely used in finance, engineering, and natural sciences. In particular, simulations indicate that algorithms based on deep learning overcome the curse of dimensionality in the numerical approximation of solutions of semilinear PDEs. For certain linear PDEs this has also been proved mathematically. The key contribution of this article is to rigorously prove this for the first time for a class of nonlinear PDEs. More precisely, we prove in the case of semilinear heat equations with gradient-independent nonlinearities that the numbers of parameters of the employed deep neural networks grow at most polynomially in both the PDE dimension and the reciprocal of the prescribed approximation accuracy. Our proof relies on recently introduced multilevel Picard approximations of semilinear PDEs.