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

TL;DR

This paper develops a nonlinear multilevel tensor-train space-time (ML-TT) framework for solving all-at-once nonlinear PDEs, integrating a coarse-to-fine TT strategy with TT-based Newton–DMRG solvers and adaptive TT-rounding. The ML-TT method computes on a hierarchy of space-time grids, prolongates coarse solutions to finer levels, and uses regularization to stabilize local TT solves, notably in advection-dominated and shock-forming regimes. Across Fisher-KPP, viscous Burgers, sine-Gordon, and KdV dynamics, ML-TT consistently achieves CT-level accuracy with significantly reduced Newton iterations and favorable wall-time scaling, while single-level TT can fail or stagnate on refined meshes. The work highlights that robust nonlinear convergence is the key bottleneck for monolithic space-time TT solvers and demonstrates that a nonlinear multilevel initialization, coupled with regularized TT-DMRG, offers a practical path toward high-fidelity, scalable simulations, with clear potential for extension to higher dimensions via TT/QTT representations.

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.

Figures (5)

• Figure 1: Traveling wave solution of Fisher-KPP equation Eq. \ref{['eqn:fkpp']}: $D=1, r=1,N_x=N_t=2^{10}, \Delta t=2/N_t$, (a) comparison with analytical solution at $t=2$, , (b) contour of the space-time solution, (c) convergence of numerical error ($\mathcal{O}(\Delta t, \Delta x^2)$) with mesh resolution $N_x, N_t$, and (d) wall-time comparison of classical time-stepping method (CT) and single- (SL) and multi-level (ML) space-time methods.
• Figure 2: Numerical solution of nonlinear viscous Burgers' equation Eq. \ref{['eqn:Burgers']} in parabolic regime: $\nu=10^{-2},a=1.01,N_x=N_t=2^{10}, \Delta t=1/N_t$, (a) comparison with analytical solution at $t=1 ~s$, (b) the space-time solution in $xt-$ plane, (c) convergence of numerical error ($\mathcal{O}(\Delta t, \Delta x^2)$) with mesh resolution $N_x, N_t$, and (d) wall-time comparison among classical time-stepping method (CT), single- (SL) and multi-level (ML) space-time methods.
• Figure 3: Evolution of a sine-wave into a shock-wave simulated by solving nonlinear viscous Burgers' equation Eq. \ref{['eqn:Burgers']}: $\nu=10^{-2},N_x=N_t=2^{10}, \Delta t=1/N_t$, (a) solutions at every 10 time step interval, (b) contour of the space-time solution.
• Figure 4: Numerical solution of the sine--Gordon equation \ref{['eqn:sg']}. The discretization uses $N_x=2^{10}$ and $N_t=2^{8}$ with time step $\Delta t=10/N_t$. (a) The numerical and analytical solutions at $t=10$, (b) space--time contour plot of the numerical solution, (c) convergence of the numerical error under mesh refinement, demonstrating first-order temporal accuracy, $\mathcal{O}(\Delta t)$, due to the use of the implicit Euler time discretization, (d) wall-clock time of the classical time-stepping method (CT) with the single-level (SL) and multilevel (ML) space--time methods.
• Figure 5: Numerical solution of the KdV equation \ref{['eqn:kdv']} for the single-soliton case. The discretization uses $N_x=N_t=2^{10}$ with time step $\Delta t=2/N_t$. (a) Comparison between the numerical and analytical solutions at $t=2$, (b) space--time contour plot of the numerical solution, (c) convergence of the numerical error with mesh refinement: second-order spatial accuracy $\mathcal{O}(\Delta x^2)$ is observed on coarse meshes, while for finer meshes the error is dominated by the first-order temporal contribution $\mathcal{O}(\Delta t)$, (d) wall-clock time of the classical time-stepping method (CT) with the single-level (SL) and multilevel (ML) space--time methods.