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Quantum-Inspired Tensor Networks for Approximating PDE Flow Maps

Nahid Binandeh Dehaghani, Ban Q. Tran, Rafal Wisniewski, Susan Mengel, A. Pedro Aguiar

TL;DR

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.

Abstract

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.

Quantum-Inspired Tensor Networks for Approximating PDE Flow Maps

TL;DR

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.

Abstract

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.
Paper Structure (27 sections, 3 theorems, 28 equations, 4 figures, 1 algorithm)

This paper contains 27 sections, 3 theorems, 28 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matrix Product States
  4. High-Dimensional States and Tensorization
  5. Matrix Product State Representation
  6. Canonical Forms
  7. Left-canonical form
  8. Right-canonical form
  9. Mixed-canonical form
  10. Matrix Product Operators
  11. Quantum-Inspired Tensor Network (QTN) Models
  12. Discretized PDE State Representation
  13. Time-Stepping Operator in Tensorized Coordinates
  14. Methods
  15. Tensor-Network Time-Stepping Pipeline
  16. ...and 12 more sections

Key Result

Proposition 1

Let $u \in \mathbb{R}^{d^n}$ be any discretized state reshaped into an order-$n$ tensor $\mathcal{U} \in \mathbb{R}^{d \times \cdots \times d}$. Then there exists an exact (i.e., without truncation) open-boundary MPS/TT representation of $\mathcal{U}$ with bond dimensions satisfying In particular, any such tensor admits an exact MPS representation with maximal bond dimension at most $d^{\lfloor n

Figures (4)

  • Figure 1: 1D advection--diffusion: (a) QTN rollout from iterating the one-step MPO predictor with truncation; (b) RK45 reference; (c) signed difference $u_{\mathrm{RK45}}-u_{\mathrm{QTN}}$ (time-aligned); (d) restart-averaged relative $\ell_2$ error vs. prediction horizon $m$.
  • Figure 2: 2D advection--diffusion: comparison of QTN rollout and RK45 reference at $t=\{0,\,0.25,\,0.50,\,0.75,\,1.00\}$. Top row: QTN rollout obtained by iterating the one-step predictor with truncation after each step (states stored in compressed MPS form). Middle row: RK45 reference solution at matching times. Bottom row: signed difference $u_{\mathrm{RK45}}-u_{\mathrm{QTN}}$.
  • Figure 3: 1D viscous Burgers (periodic): (a) QTN rollout, (b) RK45 reference, (c) signed difference $u_{\mathrm{RK45}}-u_{\mathrm{QTN}}$ (time-aligned), and (d) restart-averaged relative $\ell_2$ error versus prediction horizon $m$ (mean $\pm 1$ std).
  • Figure 4: 2D viscous Burgers: comparison of QTN rollout and RK45 reference at $t=\{0,\,0.25,\,0.50,\,0.75,\,1.00\}$. Top row: QTN rollout obtained by iterating the one-step predictor with truncation after each step. Middle row: RK45 reference snapshots at matching times. Bottom row: signed difference $u_{\mathrm{RK45}}-u_{\mathrm{QTN}}$.

Theorems & Definitions (10)

  • Remark 1: Action of an MPO on an MPS
  • Remark 2: Relevance to PDE Flow Maps
  • Remark 3
  • Proposition 1: Exact MPS/TT representation
  • proof
  • Proposition 2: Optimality of SVD-based compression
  • proof
  • Proposition 3: Multi-step error propagation
  • proof
  • Remark 4: Restart-time bound and interpretation