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Deep neural networks and other deep learning methods have very successfullybeen applied to the numerical approximation of high-dimensional nonlinearparabolic partial differential equations (PDEs), which are widely used infinance, engineering, and natural sciences. In particular, simulations indicatethat algorithms based on deep learning overcome the curse of dimensionality inthe numerical approximation of solutions of semilinear PDEs. For certain linearPDEs this has also been proved mathematically. The key contribution of thisarticle is to rigorously prove this for the first time for a class of nonlinearPDEs. More precisely, we prove in the case of semilinear heat equations withgradient-independent nonlinearities that the numbers of parameters of theemployed deep neural networks grow at most polynomially in both the PDEdimension and the reciprocal of the prescribed approximation accuracy. Ourproof relies on recently introduced multilevel Picard approximations ofsemilinear PDEs.

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