Deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear partial differential equations

Abstract

We prove that deep neural networks are capable of approximating solutions of semilinear Kolmogorov PDE in the case of gradient-independent, Lipschitz-continuous nonlinearities, while the required number of parameters in the networks grow at most polynomially in both dimension $d \in \mathbb{N}$ and prescribed reciprocal accuracy $\varepsilon$. Previously, this has only been proven in the case of semilinear heat equations.