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

TL;DR

This work develops quantum‑inspired, tensor‑network based methods for space–time PDE solving and data‑driven prediction. By encoding the joint space–time domain as a matrix product state (MPS), the authors build an all‑at‑once space–time solver with MPOs and DMRG‑like optimization, achieving accurate solutions with bond dimensions well below the full grid size. They also introduce MPS‑DMD, which leverages POD on MPS representations to produce long‑term predictions with complexity that scales logarithmically with spatial and temporal resolution, demonstrated on nonlinear problems such as Burgers’ equation and the 2D Kármán vortex street. The results show substantial data compression (often >97%) and accurate forecasts, highlighting tensor networks as a bridge between numerical PDE solvers and data‑driven modeling with potential exponential speedups under low temporal entanglement and improved interpretability through the MPS structure.

Abstract

Numerical solutions of partial differential equations (PDEs) are central to the understanding of dynamical systems. Standard approaches involving time-stepping schemes compute the solution at each time step, which becomes too costly when simulating long-term dynamics. Alternatively, space-time methods that treat the combined space-time domain simultaneously promise better stability and accuracy. Interestingly, data-driven approaches for learning and predicting dynamics, such as dynamic mode decomposition (DMD), also employ a combined space-time representation. However, the curse of dimensionality often limits the practical benefits of space-time methods. In this work, we investigate quantum-inspired methods for space-time approaches, both for solving PDEs and for making DMD 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 to make accurate long-term predictions of nonlinear systems, with runtime scaling logarithmically in both spatial and temporal resolution. This research highlights the role of tensor networks in developing effective and interpretable models, bridging the gap between numerical methods and data-driven approaches.

Figures (7)

• Figure 1: Overview of simulation and prediction of dynamical systems, using the MPS space-time encoding. Let $u(x,t)$ denote the state of the dynamical system. The evolution of the state can be simulated by solving the governing equations, given by time-dependent PDEs. Alternatively, the dynamics can be predicted by learning from the system's historical data, as represented by the time series. Using the matrix product state (MPS) encoding over the combined space-time domain, we develop the MPS counterparts of the space-time solver and dynamic mode decomposition.
• Figure 2: Entanglement entropy of the MPS space-time solution. We present the bipartite entanglement entropy as a measure of spatio-temporal correlations captured by the MPS ansatz. For linear equations, we plot the entanglement of the final solution. For nonlinear equations, we plot the intermediate iterations of the Picard scheme. For all PDEs, we use a 20-qubit MPS encoding to represent the solution on a $1024\times1024$ space-time grid. The numerical simulations are summarized in Table \ref{['tab:equations']}, and initial and boundary conditions are presented in Appendix \ref{['app:numerical_results']}. We exceptionally use the time-space ordering for the nonlinear Schrödinger equation, which is further explained in Appendix \ref{['app:NLSE']}.
• Figure 3: MPS-DMD algorithm. We assume that the snapshot matrices $X$ and $X'$ are provided in their MPS representations. First, we compute the proper orthogonal decomposition (POD) of $X$ by bringing the corresponding MPS into the mixed canonical form at the bond connecting the spatial and temporal parts. Now, the pseudoinverse of $X$, denoted by $X^+$, can be computed by inverting the singular values $\Sigma$. We then project A onto the span of $U$ to obtain the reduced evolution operator $\tilde{A}$, which is used to efficiently compute the eigen decomposition of the full operator $A$. We further use this decomposition to predict the future evolution of the system. All of the matrix operations, except for the eigen decomposition of $\tilde{A}$, are computed with only local contractions, resulting in a logarithmic scaling with both the spatial resolution and the number of snapshots.
• Figure 4: MPS-DMD prediction of nonlinear systems. (a.) 1D Burgers' equation: We start with the MPS space-time solution of Burgers' equation for a fluid with viscosity $\nu=10^{-3}$, on a $2^{10}\times2^{10}$ grid in the $(x,t)$ domain, obtained using the MPS space-time solver proposed in Sec. \ref{['sec:mps_spacetime']}. We predict the solution for the next $1000$ time steps using the MPS-DMD algorithm. We plot the behavior of the $L_2$ error of the predicted solution over time at different truncations of the space-time bond dimension, which corresponds to the number of DMD modes used. (b.) 2D Kármán vortex street phenomena: Using simulated data from Ref. dataset_cylinder, we encode the time series of the x-velocity component into the MPS space-time representation. We choose the interleaved encoding (see Ref. pisoni2025compression) for the 2D space, but maintain the space-time ordering of the MPS. We plot the x-velocity in the wake of the cylinder, which captures the vortex street phenomenon. Using only 100 DMD modes, we observe that the dominant characteristics of the velocity are recovered. We also note that even if the training error increases with time, the DMD modes generalize well for accurate predictions.
• Figure A1: Benchmarks of the MPS space-time solver. (a) We solve the 1D Heat equation $u_t-\alpha u_{xx} = 0$ with the initial condition $u(x,t=0)=\sin(x)$ on the domain $\Omega = (0,2\pi)\times(0,T]$. (b) We solve the 1D Wave equation $u_{tt}-ku_{xx}=0$ with the initial conditions given by $u(x,t=0)=\sin(\pi x)$ and $u_t(x,t=0)=0$ on the domain $\Omega = (0,10)\times(0,T]$. (c) We solve $u_{t}+u u_{x}+\nu u_{xx} =0$ with the initial condition $u(x,t=0)=\frac{2\nu\pi\sin(\pi x)}{1.01+\cos(\pi x)}$. The solution is defined on the domain $\Omega = [0,1]\times(0, T]$. (d) We solve the 1D 'focusing' case of the NLS equation $i \psi_t = - \psi_{xx} - 2 |\psi|^2 \psi$, where $\psi (x,t)$ is a complex-valued function for all $(x,t) \in [-3\pi,3\pi] \times (0,T]$. We plot the magnitude of the solution in this case. For all equations, a 20-bit MPS encodes the solution, on the domain discretized by $2^{10}\times2^{10}$ points on the $x$ and $t$ axes, respectively. The MPS space-time solution ($u_{mps}$) is presented in the corresponding top left plots, where the spatial dimension ($x$) and time ($t$) are represented along the vertical and horizontal axes, respectively. The deviation from the known analytical solution ($u^*$), i.e., error ($u_{mps}-u^*$), is plotted in the top right plots. In the bottom plots, the entanglement entropy of the MPS space-time solution is shown at each bipartition of the MPS. For nonlinear equations, the intermediate Picard iterations are specified in the entanglement plot.