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

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