A Probabilistic State Space Model for Joint Inference from Differential Equations and Data

Mechanistic models with differential equations are a key component of\nscientific applications of machine learning. Inference in such models is\nusually computationally demanding, because it involves repeatedly solving the\ndifferential equation. The main problem here is that the numerical solver is\nhard to combine with standard inference techniques. Recent work in\nprobabilistic numerics has developed a new class of solvers for ordinary\ndifferential equations (ODEs) that phrase the solution process directly in\nterms of Bayesian filtering. We here show that this allows such methods to be\ncombined very directly, with conceptual and numerical ease, with latent force\nmodels in the ODE itself. It then becomes possible to perform approximate\nBayesian inference on the latent force as well as the ODE solution in a single,\nlinear complexity pass of an extended Kalman filter / smoother - that is, at\nthe cost of computing a single ODE solution. We demonstrate the expressiveness\nand performance of the algorithm by training, among others, a non-parametric\nSIRD model on data from the COVID-19 outbreak.\n