Bayesian Identification of Nonlinear Dynamics (BINDy)

The Sparse Identification of Nonlinear Dynamics (SINDy) framework has been shown to be effective in learning interpretable and parsimonious models directly from data. However, existing SINDy derivatives can be computationally expensive and may struggle to learn the correct model equations from noisy and small datasets.

We propose a Bayesian extension to SINDy for learning sparse equations from data. Our method shows more robust capability in learning the correct model in the low-data limit, as it sparsifies the model based on both the value and the distribution of the parameters during regression, and uses the marginal likelihood (evidence) to rank and select the candidate models. The proposed method uses Laplace's method to approximate the Bayesian likelihood and evidence, avoiding the need for computationally expensive Markov chain Monte Carlo (MCMC) sampling. This results in a significant speedup in computation compared to other Bayesian SINDy methods, while still achieving comparable or better performance than existing methods such as Ensemble-SINDy.

We demonstrate the effectiveness of the proposed method on a variety of problems, including learning the Lotka-Volterra equations from 21 experimental data points of the Hudson Bay Lynx-Hare population dataset, and the Lorenz system using tens of noisy data points. We also apply the method to a real-life dataset in fluids, such as the measurements of the quasi-2D Kolmogorov-like flow, showing how it can learn both the high-order equations and low-order dynamics with just a tiny subset of the data.