PNNL

Abstract

We propose a probabilistic model discovery method for identifying ordinary differential equations (ODEs) governing the dynamics of observed multivariate data. Our method is based on the "sparse identification of nonlinear dynamics" (SINDy) framework, in which target ODE models are expressed as a sparse linear combinations of prespecified candidate functions. Promoting parsimony through sparsity in SINDy leads to interpretable models that generalize well to unknown data. Instead of targeting point estimates of the SINDy (linear combination) coefficients, in this work we estimate these coefficients via sparse Bayesian inference. The resulting method, UQ-SINDy, quantifies not only the uncertainty in the values of the SINDy coefficients due to observation errors and limited data, but also the probability of inclusion of each candidate function in the linear combination. UQ-SINDy promotes robustness against observation noise and limited data, interpretability (in terms of model selection and inclusion probabilities), and generalization capacity for out-of-sample forecast. We perform sparse inference in UQ-SINDy employing Markov Chain Monte Carlo, and we explore two sparsifying priors: The spike-and-slab prior, and the regularized horseshoe prior. We apply UQ-SINDy to synthetic nonlinear datasets from a Lotka-Volterra model and a nonlinear oscillator, and to a real world dataset of lynx and hare populations. We find that UQ-SINDy is able to discover accurate and meaningful models even in the presence of noise and sparse samples.