Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in L^p-sense

2024/09/30 by Neufeld, Ariel, Nguyen, Tuan Anh · 1 citation
#FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Numerical Analysis (math.NA) #Probability (math.PR)

paper · doi:10.48550/arxiv.2409.20431

Abstract

We prove that multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation are capable of approximating solutions of semilinear Kolmogorov PDEs in L^\mathfrakp-sense, \mathfrakp∈ [2,∞), in the case of gradient-independent, Lipschitz-continuous nonlinearities, while the computational effort of the multilevel Picard approximations and the required number of parameters in the neural networks grow at most polynomially in both dimension d∈ ℕ and reciprocal of the prescribed accuracy ε.

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