Combining Monte Carlo and Tensor-network Methods for Partial Differential Equations via Sketching
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909955306553344 |
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| author | Chen, Yian Khoo, Yuehaw Yu, Ziang |
| author_facet | Chen, Yian Khoo, Yuehaw Yu, Ziang |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_17884 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Combining Monte Carlo and Tensor-network Methods for Partial Differential Equations via Sketching Chen, Yian Khoo, Yuehaw Yu, Ziang Numerical Analysis Machine Learning 15A69, 62G07, 65M75, 65M22, 82B05, 82B10 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. |
| title | Combining Monte Carlo and Tensor-network Methods for Partial Differential Equations via Sketching |
| topic | Numerical Analysis Machine Learning 15A69, 62G07, 65M75, 65M22, 82B05, 82B10 |
| url | https://arxiv.org/abs/2305.17884 |