Variational Monte Carlo - Bridging between Numerics and Statistical Learning

For solving solve high-dimensional PDE's which can be casted into a variational framework restricted to appropriate possibly non-linear and even non-convex model classes. In Variational Monte Carlo we replace our objective functional by an empirical functional, in a similar way as for risk minimization or loss functions in statistical learning. For the optimization we need only computable gradients at sample points.At a price, we consider {\em convergence in probability}, i.e. error estimates holds with high probability. The analysis is carried out in the spirit of Cucker and Smale, and first estimates about covering numbers for hierarchical tensors are available. As a model class we consider hierarchical tensors (HT) mainly in TT form.

This approach has been carried out earlier for parametric problems in uncertainty quantifcation (with M. Eigel, P. Trunschke and S. Wolf). We present in this talk an application to approximate the meta-stable eigenfunctions of the corresponding Backward Kolmogorov operator by numerical approximation of the transfer operator (also know as Koopman operator), and vice versa the Fokker Planck operator (with K. Fackeldey, F. N\"uske and F. Noe). We further consider the approximation of the value function for a closed loop bilinear control problem, described by the Hamilton Jacobi equation. The policy iteration from dynamic programming consists in alternating between updating the optimal feedback law and the solution of an inhomogneous Backward Kolmogorov equation. We solve the latter by means of HT/TT tensors and variational Monte Carlo. First numerical results are presented (with L. Sallandt and M. Oster).