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We propose a probabilistic model discovery method for identifying ordinarydifferential equations (ODEs) governing the dynamics of observed multivariatedata. Our method is based on the sparse identification of nonlinear dynamics(SINDy) framework, in which target ODE models are expressed as a sparse linearcombinations of pre-specified candidate functions. Promoting parsimony throughsparsity in SINDy leads to interpretable models that generalize to unknowndata. Instead of targeting point estimates of the SINDy (linear combination)coefficients, in this work we estimate these coefficients via sparse Bayesianinference. The resulting method, uncertainty quantification SINDy (UQ-SINDy),quantifies not only the uncertainty in the values of the SINDy coefficients dueto observation errors and limited data, but also the probability of inclusionof each candidate function in the linear combination. UQ-SINDy promotesrobustness against observation noise and limited data, interpretability (interms of model selection and inclusion probabilities), and generalizationcapacity for out-of-sample forecast. Sparse inference for UQ-SINDy employsMarkov Chain Monte Carlo, and we explore two sparsifying priors: thespike-and-slab prior, and the regularized horseshoe prior. We apply UQ-SINDy tosynthetic nonlinear data sets from a Lotka-Volterra model and a nonlinearoscillator, and to a real-world data set of lynx and hare populations. We findthat UQ-SINDy is able to discover accurate and meaningful models even in thepresence of noise and limited data samples.

Sparsifying Priors for Bayesian Uncertainty Quantification in Model Discovery