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

arXiv: https://arxiv.org/abs/2305.17884 · HTML: https://mlsanity.com/paper/2305.17884/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 54 references that are in the map.
- First posted on arXiv 29 May 2023.

## Builds on (most cited first, 8 of 8)

- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [An Introduction to Deep Generative Modeling](https://mlsanity.com/paper/2103.05180/) (2021)
- [Fast and Guaranteed Tensor Decomposition via Sketching](https://mlsanity.com/paper/1506.04448/) (2015)
- [Generative modeling via tensor train sketching](https://mlsanity.com/paper/2202.11788/) (2022)
- [Generative Modeling via Hierarchical Tensor Sketching](https://mlsanity.com/paper/2304.05305/) (2023)
- [Generative Modeling via Tree Tensor Network States](https://mlsanity.com/paper/2209.01341/) (2022)
- [High-dimensional density estimation with tensorizing flow](https://mlsanity.com/paper/2212.00759/) (2022)

## Abstract (arXiv)

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.