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
<|reference_start|>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: 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 $L^\mathfrak{p}$-sense, $\mathfrak{p}\in [2,\infty)$, 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 $d\in \mathbb{N}$ and reciprocal of the prescribed accuracy $\epsilon$.<|reference_end|>
@article{neufeld2024multilevel,
    title={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},
    author={Ariel Neufeld, Tuan Anh Nguyen},
    journal={arXiv preprint arXiv:2409.20431},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.20431},
    primaryClass={math.NA cs.LG cs.NA math.PR}
}