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

TL;DR

This paper presents a general framework that merges Monte Carlo simulations with tensor-network representations (MPS/TT) for solving high-dimensional PDEs. It leverages a tensor-network sketching technique to reconstruct low-rank TT representations from particle samples, enabling efficient application of semigroups and variance control. The authors provide convergence guarantees for the Fokker-Planck equation and demonstrate the approach on quantum imaginary-time evolution (via AFQMC) and overdamped Langevin dynamics, achieving accurate ground-state energies and density evolution in high-dimensional settings. The work offers a scalable alternative to traditional methods by exploiting low-rank tensor structures and probabilistic sampling, with potential impact across statistical mechanics and quantum many-body simulations.

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.

Figures (11)

• Figure 2.1: Tensor diagram for a $d$-dimensional MPS/TT and MPO. (a): An MPS/TT representing $u$ in Definition \ref{['def:mps']}. The exposed legs indicates the MPS/TT takes $x_1,\ldots,x_d$ as inputs, and the connected legs indicate the summation over $\alpha_1,\ldots,\alpha_{d-1}$. (b) An MPO representing $O$. Each tensor core has two exposed legs pointing upwards and downwards, respectively, indicating two free dimensions.
• Figure 2.2: Tensor diagrams to reduce the bond dimensions of MPS/TT via truncated SVD. The regrouping operation for each reduced tensor $\bar{\mathcal{G}}_k$ is highlighted by red dashed boxes.
• Figure 3.1: Tensor diagram for the workflow of estimating an MPS/TT from particles. Step (a) shows how $\phi_t$ is represented as empirical distribution $\hat{\phi}_{t+1}$. Step (b), (c), (d) shows how to form $A_k[\hat{\phi}_{t+1}], B_k[\hat{\phi}_{t+1}]$ in \ref{['eq:sampled A B']} and use them to solve for $\mathcal{G}_k$. Here we use the determination of $\mathcal{G}_k$ where $k=3$ as an example.
• Figure 4.1: Tensor diagram for approximating evolution equation as MPO-MPS products. We remove the internal legs connecting the tensor cores in MPO to indicate the MPO has rank $1$.
• Figure 6.1: Example of $2$D space-filling curve for Ising model of $4\times 4$ lattices.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Definition 3
• Remark 1: Rank of the MPS/TT representation
• Remark 2
• Lemma 1: Theorem 3.48, wendland2017numerical
• Lemma 2
• proof
• Definition 4
• Lemma 3