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Deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear partial differential equations
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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.
Forward citations
Cited by 3 Pith papers
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Deep neural networks can provably solve Bellman equations for Markov decision processes without the curse of dimensionality
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A brief review of the Deep BSDE method for solving high-dimensional partial differential equations
A review by the method's inventors that describes the Deep BSDE approach to high-dimensional PDEs and surveys follow-up work.
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