BCAM Scientific Seminar: From Monte Carlo to neural networks approximations of boundary value problems

DATE: 20 September
LOCATION: BCAM Seminar Room and online
TIME: 16:00 | Please, note that at 15:30 will start the coffee break.

Abstract:
We present probabilistic and neural network approximations for solutions to Poisson equation subject to H" older continuous Dirichlet boundary conditions in general bounded domains in $mathbb{R}^d$. Our main results are two-folded: On the one hand we show that the solution to Poisson equation can be numerically approximated in the sup-norm by Monte Carlo methods, without the curse of high dimensions and efficiently with respect to the prescribed approximation error. The proposed approach reveals that probabilistic representations in conjunction with Monte Carlo methods are globally efficient to solve elliptic partial differential equations, in the sense that the random samples required by Monte Carlo do not depend on the location where the solution needs to be approximated. On the other hand, we show that the obtained Monte Carlo solver renders ReLU deep neural network (DNN) solutions to Poisson problem, whose sizes depend at most polynomially on $d$ and on the desired error. A byproduct of the herein proposed Monte Carlo solver is that it can be easily used to sample the corresponding DNNs, whilst the theoretical error estimates can be used to ensure that such sampled DNNs obey a prescribed error with high probability.


You can follow the BCAM Scientific Seminar online at this link https://us06web.zoom.us/j/83918020101?pwd=bXF5bUExcGVJUzRLeWhsTGhvNGhwZz09