Parallel-in-Time Probabilistic Solutions for Time-Dependent Nonlinear Partial Differential Equations

Abstract

We present an efficient probabilistic solver for time-dependent nonlinear partial differential equations. We formulate our method as the maximum a posteriori solver for a constrained risk problem on a reproducing kernel Hilbert space induced by a spatiotemporal Gaussian process prior. We show that for a suitable choice of temporal kernels, the risk objective can be minimized efficiently via a Gauss-Newton algorithm corresponding to an iterated extended Kalman smoother (IEKS). Furthermore, by leveraging a parallel-in-time implementation of IEKS, our algorithm can take advantage of massively parallel graphical processing units to achieve logarithmic instead of linear scaling with time. We validate our method numerically on popular benchmark problems.