  Inverse problems with spatiotemporal observations are ubiquitous in
scientific studies and engineering applications. In these spatiotemporal
inverse problems, observed multivariate time series are used to infer
parameters of physical or biological interests. Traditional solutions for these
problems often ignore the spatial or temporal correlations in the data (static
model), or simply model the data summarized over time (time-averaged model). In
either case, the data information that contains the spatiotemporal interactions
is not fully utilized for parameter learning, which leads to insufficient
modeling in these problems. In this paper, we apply Bayesian models based on
spatiotemporal Gaussian processes (STGP) to the inverse problems with
spatiotemporal data and show that the spatial and temporal information provides
more effective parameter estimation and uncertainty quantification (UQ). We
demonstrate the merit of Bayesian spatiotemporal modeling for inverse problems
compared with traditional static and time-averaged approaches using a
time-dependent advection-diffusion partial different equation (PDE) and three
chaotic ordinary differential equations (ODE). We also provide theoretic
justification for the superiority of spatiotemporal modeling to fit the
trajectories even it appears cumbersome (e.g. for chaotic dynamics).
