Statistical mechanics of dynamical system identification*

Recovering the dynamical equations from observed noisy trajectory data constitutes the problem of system identification. Approaches such as Sparse Identification of Nonlinear Dynamics (SINDy) are usually framed as minimizing a loss function that balances the data fit with parsimonious regularization, requiring a trial-and-error selection of hyperparameters. In this work we formulate system identification as a two-level Bayesian inference problem that explicitly separates variable selection from variable values and uses statistical mechanics techniques to compute the posterior parameter distribution in closed form and avoid Monte Carlo sampling. The low-data limit of this approach provides an uncertainty quantification of the identified models. The high-data limit resembles the thermodynamic limit and leads to a sharp noise-induced phase transition between correct and incorrect identification. The statistical mechanics perspective can be integrated with other SINDy variants and applied to sparse regression problems in other contexts.