For many high dimensional PDEs of practical interest, e.g. Back-ward Kolmogorov equations etc. the PDE operator cannot be easily expanded in matrix product operator form. In this case, we propose a variational Monte Carlo approach con?ned to the manifold of tree based tensors with ?xed multi-rank (i.e. bond dimensions).
In particular the (stochastic) HJB can be reformulated by an (un-coupled) Forward Backward SDE system. The forward dynamics can be computed easily by standard Euler-Mayurana scheme. For the backward equation for the value function, we use variational inter-polation (Bender et al.) by solving a regression problem in each time step. For this purpose we use tree based tensor networks, in partic-ular MPS/TT. The forward backward SDE is linked with (parabolic) PDEs by a non-linear Feynman-Kac theorem. This o?ers the possi-bility to computed approximate (local) solutions of wide class of non-linear parabolic PDE’s. for our solving our HJB equations, we need to compute the gradient of the value functions with desired precision. Solving regression problems by means of HT/TT tensors with good approximation of the gradients requires additional attention, and has been the technical key for a successful treatment. First numerical re-sults has been obtained by my PhD students Leon Sallandt and M. Oster. We have compared with already published results using deep neural networks (E & Jentzen et al., Pham et al.). All results we have found in literature could be con?rmed by our approach or even improved. Large part of the numerical method can be transfered for-mally for deterministic optimal control, and works well if additional regularity of the solution is granted."
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