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

Yian Chen; Yuehaw Khoo; Ziang Yu

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arXiv · 2305.17884

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

Yian Chen·Yuehaw Khoo·Ziang Yu

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Abstract

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.

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Yian Chen, Yuehaw Khoo, Ziang Yu. 2023-05-29. Combining Monte Carlo and Tensor-network Methods for Partial Differential Equations via Sketching. https://arxiv.org/abs/2305.17884

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