Modeling and inference of the dynamics of spatiotemporally evolving systems using evolving gaussian processes

In this work, we present a new differentially-constrained machine learning model, termed Evolving Gaussian Processes (E-GP), for modeling and inference of spatiotemporally evolving dynamical systems. We show that an E-GP model can be used to estimate the latent state of large-scale physical systems of this type, and furthermore that a single E-GP model can generalize over multiple physically-similar systems over a range of parameters using only a few training sets. It is also shown that an E-GP model provides access to practical physical insights into the dynamic structure of the system(s) it is trained on. In particular, from spectral analysis of the linear dynamic layer in the top level of the E-GP model, one may derive the Koopman modes and eigenvalues of the system. We are also able to derive the spatial distribution of the invariant subspaces of a system using a new clustering method. This information can be used for sensor placement and/or mobile agent path planning for robust inference of the state of the system using few measurements. We primarily demonstrate our method on computational flow dynamics (CFD) data sets on fluid flowing past a cylinder at different Reynolds numbers. Though these systems are governed by highly nonlinear partial differential equations (the Navier-Stokes equations), we show that their major dynamical modes can be captured by a linear dynamical layer over the temporal evolution of the weights of stationary kernels