Bayesian tree-adjoining grammars improve uncertainty in dynamic equation discovery
Equation discovery with Bayesian tree-adjoining grammars
Machine Learning
Summary
Finding mathematical equations to describe complex changing systems can be tricky and uncertain. The authors use a method called Tree-Adjoining Grammars combined with Bayesian statistics to better capture this uncertainty and suggest possible models. Their approach can include known physics knowledge, improving predictions for real-world data like wave forces. This method helps understand what equations best explain system behaviors while showing how confident we can be in those models.
What this means in practice
- •For control system engineers: Build models that better quantify uncertainty when discovering nonlinear system equations from data, improving control design reliability.
- •For marine engineers: Enhance prediction of wave loading effects on structures by embedding physics-informed models with uncertainty estimates from sensor data.
Authors
Christopher A. Lindley, Nikolaos Dervilis, Keith Worden
Abstract
Tree-Adjoining Grammars (TAGs) have recently been introduced to Nonlinear System Identification (NLSI) as a means of encoding an entire model class as a finite set of grammatical rules, from which candidate models are assembled as trees. Existing TAG-based identifiers rely on evolutionary optimisation and return point estimates of the model structure. This paper instead proposes the TAG framework within a Bayesian setting. A generative prior is defined over tree structures and their parameters, and a Reversible-Jump MCMC sampler with structure-preserving tree moves is used to infer the joint posterior over model structure, parameters and predictions. Two training objectives are considered; that is, a one-step-ahead objective with conjugate parameter proposals, and a simulation-based objective handled by likelihood-free inference. The approach is validated on a simulated polynomial NARX system, the Silverbox benchmark, and wave-loading data from the Christchurch Bay Tower, where embedding Morison's equation as a fixed initial tree yields a grey-box model that outperforms the physics-driven baseline. The results demonstrate that Bayesian TAGs are well suited to quantifying uncertainty in equation discovery for dynamical systems and to fitting physics-informed models.