Combining Particle and Tensor-network Methods for Partial Differential Equations via Sketching

In this paper, we propose a general framework for solving high-dimensional partial differential equations with tensor networks. Our approach offers a comprehensive solution methodology, wherein we employ a combination of particle simulations to update the solution and re-estimations of the new solution as a tensor-network using a recently proposed tensor train sketching technique. Our method can also be interpreted as an alternative approach for performing particle number control by assuming the particles originate from an underlying tensor network. We demonstrate 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.