Fast and Flexible Quantum-Inspired Differential Equation Solvers with Data Integration

TL;DR

This paper introduces a quantum-inspired tensor train method for solving high-dimensional PDEs efficiently, achieving logarithmic scaling and integrating data-driven learning to improve accuracy and reduce training time.

Contribution

It presents a novel QTT-based approach for PDEs that combines quantum-inspired techniques with neural networks for enhanced efficiency and accuracy.

Findings

Logarithmic scaling in memory and computational cost for PDE solutions

Effective handling of both linear and nonlinear PDEs

Reduced training time through data-driven learning

Abstract

Accurately solving high-dimensional partial differential equations (PDEs) remains a central challenge in computational mathematics. Traditional numerical methods, while effective in low-dimensional settings or on coarse grids, often struggle to deliver the precision required in practical applications. Recent machine learning-based approaches offer flexibility but frequently fall short in terms of accuracy and reliability, particularly in industrial contexts. In this work, we explore a quantum-inspired method based on quantized tensor trains (QTT), enabling efficient and accurate solutions to PDEs in a variety of challenging scenarios. Through several representative examples, we demonstrate that the QTT approach can achieve logarithmic scaling in both memory and computational cost for linear and nonlinear PDEs. Additionally, we introduce a novel technique for data-driven learning within the quantum-inspired framework, combining the adaptability of neural networks with enhanced accuracy and reduced training time.