Quantum-Inspired Tensor Networks for Approximating PDE Flow Maps
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911453483630592 |
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| author | Dehaghani, Nahid Binandeh Tran, Ban Q. Wisniewski, Rafal Mengel, Susan Aguiar, A. Pedro |
| author_facet | Dehaghani, Nahid Binandeh Tran, Ban Q. Wisniewski, Rafal Mengel, Susan Aguiar, A. Pedro |
| contents | We investigate quantum-inspired tensor networks (QTNs) for approximating flow maps of hydrodynamic partial differential equations (PDEs). Motivated by the effective low-rank structure that emerges after tensorization of discretized transport and diffusion dynamics, we encode PDE states as matrix product states (MPS) and represent the evolution operator as a structured low-rank matrix product operator (MPO) in tensor-train form (e.g., arising from finite-difference discretizations assembled in MPO form). The MPO is applied directly in MPS form, and rank growth is controlled via canonicalization and SVD-based truncation after each step. We provide theoretical context through standard matrix product properties, including exact MPS representability bounds, local optimality of SVD truncation, and a Lipschitz-type multi-step error propagation estimate. Experiments on one- and two-dimensional linear advection-diffusion and nonlinear viscous Burgers equations demonstrate accurate short-horizon prediction, favorable scaling in smooth diffusive regimes, and error growth in nonlinear multi-step predictions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_15906 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantum-Inspired Tensor Networks for Approximating PDE Flow Maps Dehaghani, Nahid Binandeh Tran, Ban Q. Wisniewski, Rafal Mengel, Susan Aguiar, A. Pedro Numerical Analysis Quantum Physics We investigate quantum-inspired tensor networks (QTNs) for approximating flow maps of hydrodynamic partial differential equations (PDEs). Motivated by the effective low-rank structure that emerges after tensorization of discretized transport and diffusion dynamics, we encode PDE states as matrix product states (MPS) and represent the evolution operator as a structured low-rank matrix product operator (MPO) in tensor-train form (e.g., arising from finite-difference discretizations assembled in MPO form). The MPO is applied directly in MPS form, and rank growth is controlled via canonicalization and SVD-based truncation after each step. We provide theoretical context through standard matrix product properties, including exact MPS representability bounds, local optimality of SVD truncation, and a Lipschitz-type multi-step error propagation estimate. Experiments on one- and two-dimensional linear advection-diffusion and nonlinear viscous Burgers equations demonstrate accurate short-horizon prediction, favorable scaling in smooth diffusive regimes, and error growth in nonlinear multi-step predictions. |
| title | Quantum-Inspired Tensor Networks for Approximating PDE Flow Maps |
| topic | Numerical Analysis Quantum Physics |
| url | https://arxiv.org/abs/2602.15906 |