A quantum-inspired multi-level tensor-train monolithic space-time method for nonlinear PDEs

TL;DR

This paper introduces a multilevel tensor-train framework for solving nonlinear PDEs in a space-time formulation, improving convergence and efficiency over traditional methods, especially in complex nonlinear regimes.

Contribution

It develops a multilevel TT approach with transfer operators and adaptive-rank algorithms, enhancing robustness and accuracy in high-dimensional nonlinear PDE simulations.

Findings

Multilevel TT converges where single-level fails.

Significant reduction in computational cost for high-fidelity simulations.

Effective across diverse nonlinear PDEs including Fisher-KPP, Burgers, sine-Gordon, and KdV.

Abstract

We propose a multilevel tensor-train (TT) framework for solving nonlinear partial differential equations (PDEs) in a global space-time formulation. While space-time TT solvers have demonstrated significant potential for compressed high-dimensional simulations, the literature contains few systematic comparisons with classical time-stepping methods, limited error convergence analyses, and little quantitative assessment of the impact of TT rounding on numerical accuracy. Likewise, existing studies fail to demonstrate performance across a diverse set of PDEs and parameter ranges. In practice, monolithic Newton iterations may stagnate or fail to converge in strongly nonlinear, stiff, or advection-dominated regimes, where poor initial guesses and severely ill-conditioned space-time Jacobians hinder robust convergence. We overcome this limitation by introducing a coarse-to-fine multilevel strategy fully embedded within the TT format. Each level refines both spatial and temporal resolutions while transferring the TT solution through low-rank prolongation operators, providing robust initializations for successive Newton solves. Residuals, Jacobians, and transfer operators are represented directly in TT and solved with the adaptive-rank DMRG algorithm. Numerical experiments for a selection of nonlinear PDEs including Fisher-KPP, viscous Burgers, sine-Gordon, and KdV cover diffusive, convective, and dispersive dynamics, demonstrating that the multilevel TT approach consistently converges where single-level space-time Newton iterations fail. In dynamic, advection-dominated (nonlinear) scenarios, multilevel TT surpasses single-level TT, achieving high accuracy with significantly reduced computational cost, specifically when high-fidelity numerical simulation is required.