Accurate and efficient data assimilation models to track, predict, and control the behavior of complex dynamical systems in real time have become a vital and necessary component of digital twins. Inverse problems underlying these models require reconstructing spatiotemporal models that manifest sharp spatial features with edges and evolving temporal structures. In such examples it is worth mentioning dynamic medical imaging or temperature estimation and prediction in evolving physical systems.

Classical Gaussian process priors tend to oversmooth sharp and edge-preserving features, limiting their use in high-resolution or applications with edge-preserving desired reconstructions. In this talk, we discuss a spatiotemporal Besov process (STBP) -- a novel class of Bayesian priors that combines the spatial adaptability of Besov processes with temporally correlated stochastic structures governed by Q-exponential processes. Such formulation captures spatial inhomogeneity while maintaining realistic temporal dynamics.

Further, we discuss the mathematical properties of STBP. Its performance is illustrated through examples, including dynamic CT imaging, spatiotemporal temperature inference, and a challenging Navier–Stokes inverse problem. The results highlight how this approach can enhance the fidelity and uncertainty quantification of digital twin models where data are incomplete, noisy, and/or dynamically evolving.

This is based on joint work with (Shiwei Lan, Shuyi Li, Weining Shen) and (Jonathan Lindbloom, Jan Glaubitz, Youssef Marzouk).