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

Understand

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.