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

Lucas Arenstein, Martin Mikkelsen, Michael Kastoryano

TL;DR

The paper introduces a quantum-inspired framework based on quantized tensor trains (QTT) to solve high-dimensional PDEs with strong multiscale features by compressing operators and solutions into low-rank representations. It develops space-time QTT formulations and ALS/MALS-based solvers, enabling logarithmic scaling ${\mathcal{O}}(\log(NT))$ in spatial and temporal degrees of freedom, and demonstrates substantial speedups over traditional solvers and PINNs on Poisson and Burgers' equations. A data-driven extension embeds observations via spline interpolation into the QTT framework, enabling learning from data without retraining while preserving accuracy. The approach shows promise for high-dimensional applications (e.g., finance, kinetic equations, turbulence) and offers a pathway toward rigorous error analysis and bond-dimension management in nonlinear, time-dependent PDEs.

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.

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

TL;DR

The paper introduces a quantum-inspired framework based on quantized tensor trains (QTT) to solve high-dimensional PDEs with strong multiscale features by compressing operators and solutions into low-rank representations. It develops space-time QTT formulations and ALS/MALS-based solvers, enabling logarithmic scaling in spatial and temporal degrees of freedom, and demonstrates substantial speedups over traditional solvers and PINNs on Poisson and Burgers' equations. A data-driven extension embeds observations via spline interpolation into the QTT framework, enabling learning from data without retraining while preserving accuracy. The approach shows promise for high-dimensional applications (e.g., finance, kinetic equations, turbulence) and offers a pathway toward rigorous error analysis and bond-dimension management in nonlinear, time-dependent PDEs.

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.

Paper Structure

This paper contains 37 sections, 1 theorem, 70 equations, 7 figures, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Outline
  3. Tensor Networks
  4. The tensor train
  5. Graphical notation
  6. The quantized tensor train
  7. Low-rank QTT Representation of a Function
  8. Explicit Analytic Low-Rank QTT Representations
  9. TT-SVD
  10. Tensor cross interpolation
  11. Multiscale Interpolative QTT Construction
  12. Finite Difference in QTT format
  13. Solving a system of linear equations - ALS
  14. Finite Difference Method for the 2D Poisson in QTT Format
  15. Scaling Comparison: QTT Solver vs. Algebraic Multigrid
  16. ...and 22 more sections

Key Result

Lemma 1

Let $I = \left(\right)$, $J = \left(\right)$, $J' = \left(\right)$, and $\alpha, \beta, \gamma\in \mathbb{C}$, then for any integer $c \geq 2$, the $2^c \times 2^c$ matrix of size has an explicit QTT representation with bond dimension $3$, given by:

Figures (7)

  • Figure 1: Log-log plot comparing the run time and accuracy between PyAMG and QTT when a low rank representation of the boundary term is available.
  • Figure 2: MSE of the space-time QTT solver as a function of the number of runs. The runs to convergence is essentially independent of the parameters of the PDE.
  • Figure 3: QTT interpolation on 7 data points sampled from a function $f(x)=\sin(3x)^2+\cos(5x)^3$. We used 8 cores and 25 interpolation nodes.
  • Figure 4: Point-wise absolute error between the QTT and the analytical solution, with an average MSE of order $10^{-6}$.
  • Figure 5: Solution of the 2D heat equation at different time steps.
  • ...and 2 more figures

Theorems & Definitions (1)

  • Lemma 1