Physics-Informed Functional Tucker Method with RKHS Factors for Sparse Spatiotemporal Reconstruction

Reconstructing physical dynamics from sparse and irregular spatiotemporal observations is a fundamental challenge in scientific research. However, sparsity leaves large regions of the field unconstrained, making the inverse problem ill-posed: a model may fit the observed sensors while producing nonphysical off-sensor structures or unreliable predictions between support times. To address this challenge, we propose PhysFTM---a physics-informed functional Tucker method that parameterizes Tucker mode factors in reproducing kernel Hilbert space (RKHS) and enforces governing physical laws through finite-difference residuals at collocation points. These physical constraints are essential for credible reconstruction, since most query locations are never directly supervised by data. Specifically, PhysFTM first reconstructs continuous spatial fields at observed times via an alternating minimization scheme, with separate treatments for the Tucker core tensor and the RKHS-based factors. To further enable continuous temporal resolution, PhysFTM constructs a continuous trajectory in Tucker-core space, which is decoded by the learned spatial RKHS-FTM representation to recover the full spatiotemporal field at arbitrary query times. Experiments on Allen--Cahn and Navier--Stokes with varying observation ratios demonstrate that PhysFTM achieves superior reconstruction accuracy, improved physical consistency, and stronger continuous-time modeling capability.