arXiv2026
The numerical solution of time-dependent PDEs posed over the combined space-time domain, instead of treating time sequentially, leads to space-time methods. Treating time as another spatial dimension allows parallelization across time and simultaneous adaptivity in spatial and temporal discretizations. However, space-time formulations suffer from the curse of dimensionality due to the (d+1)-dimensional domain. Time-series forecasting methods, such as dynamic mode decomposition (DMD), also suffer the same scaling for storing and processing measurement data. In this work, we investigate whether quantum-inspired tensor network methods can mitigate this curse for both solving PDEs and data-driven modeling. We formulate a space-time matrix product state (MPS) encoding to represent either the space-time solution or time-series data. Using this representation, we develop a unified framework that combines an MPS space-time PDE solver with an MPS-DMD algorithm. We benchmark our MPS space-time solver for linear and nonlinear PDEs in both (1+1) and (2+1) dimensions, including non-diffusive regimes such as shockwave formation and Allen-Cahn phase separation. We observe that the MPS ansatz efficiently captures the spatio-temporal correlations of the solutions, with a 99% reduction in parameters with respect to the grid size. We demonstrate the MPS-DMD algorithm in two cases. First, the MPS space-time solver generates the MPS encoding of the time-series data. We then apply MPS-DMD directly to predict the dynamics without solving PDEs. Second, for the general case where data is available as dense arrays, MPS-DMD needs an intermediate compression step. We implement a streaming-like compression algorithm and observe a 97% reduction in memory, with MPS-DMD predictions matching those of exact DMD.