Efficient Bayesian inference for large chaotic dynamicalsystems

Abstract. Estimating parameters of chaotic geophysical models is challenging due to these models' inherent unpredictability. With temporally sparse long-range observations, these models cannot be calibrated using standard least squares or filtering methods. Obvious remedies, such as averaging over temporal and spatial data to characterize the mean behavior, do not capture the subtleties of the underlying dynamics. We perform Bayesian inference of parameters in high-dimensional and computationally demanding chaotic dynamical systems by combining two approaches: (i) measuring model-data mismatch by comparing chaotic attractors, and (ii) mitigating the computational cost of inference by using surrogate models. Specifically, we construct a likelihood function suited to chaotic models by evaluating a distribution over distances between points in the phase space; this distribution defines a summary statistic that depends on the attractor's geometry, rather than on pointwise matching of trajectories. This statistic is computationally expensive to simulate, compounding the usual challenges of Bayesian computation with physical models. Thus we develop an inexpensive surrogate for the log-likelihood via local approximation Markov chain Monte Carlo, which in our simulations reduces the time required for accurate inference by orders of magnitude. We investigate the behavior of the resulting algorithm on model problems, and then use a quasi-geostrophic model to demonstrate its large-scale application.