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
This work addresses the curse of dimensionality in solving high-dimensional semilinear Kolmogorov PDEs, which is a significant problem for researchers and practitioners in numerical analysis and scientific computing.
This paper demonstrates that multilevel Picard approximations and deep neural networks with specific activation functions (ReLU, leaky ReLU, softplus) can approximate solutions of semilinear Kolmogorov PDEs in an $L^p$-sense. The computational effort for Picard approximations and the number of neural network parameters grow polynomially with dimension and the reciprocal of the desired accuracy, effectively overcoming the curse of dimensionality.
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 $ε$.