  In this paper, we propose a general framework for solving high-dimensional
partial differential equations with tensor networks. Our approach uses
Monte-Carlo simulations to update the solution and re-estimates the new
solution from samples as a tensor-network using a recently proposed tensor
train sketching technique. We showcase the versatility and flexibility of our
approach by applying it to two specific scenarios: simulating the Fokker-Planck
equation through Langevin dynamics and quantum imaginary time evolution via
auxiliary-field quantum Monte Carlo. We also provide convergence guarantees and
numerical experiments to demonstrate the efficacy of the proposed method.
