Quantum-inspired space-time PDE solver and dynamic mode decomposition

TL;DR

This paper introduces a quantum-inspired tensor network method using matrix product states to efficiently solve PDEs and predict long-term dynamics, reducing computational complexity and capturing spatio-temporal correlations.

Contribution

It presents a novel MPS-based space-time PDE solver and an MPS-DMD algorithm for long-term predictions, combining numerical and data-driven approaches with improved efficiency.

Findings

MPS ansatz accurately captures spatio-temporal correlations

Runtime scales logarithmically with resolution

Method enables cheap, accurate long-term predictions

Abstract

The curse of dimensionality is ubiquitous in both numerical and data-driven methods. This is particularly severe for space-time methods, which treat the combined space-time domain simultaneously. We investigate the effectiveness of a quantum-inspired approach in alleviating this curse, both for solving PDEs and making data-driven predictions. We achieve this goal by treating both spatial and temporal dimensions within a single matrix product state (MPS) encoding. First, we benchmark our MPS space-time solver for both linear and nonlinear PDEs, observing that the MPS ansatz accurately captures the underlying spatio-temporal correlations while having significantly fewer degrees of freedom. Second, we develop an MPS-DMD algorithm for accurate long-term predictions of nonlinear systems, with runtime scaling logarithmically with both spatial and temporal resolution. We also demonstrate an application where both methods can be combined for cheap and accurate prediction of long-term dynamics. This research highlights the role of tensor networks in developing effective, interpretable models that bridge the gap between numerical methods and data-driven approaches.