Equation discovery with Bayesian tree-adjoining grammars

2026-09-25

AI Summary

This study addresses the inability of point estimation to quantify model structural uncertainty in nonlinear system identification by proposing, for the first time, a Bayesian Tree Adjoining Grammar (TAG) framework. This approach introduces TAG into a Bayesian paradigm, jointly defining tree structures and parameters through generative priors. By combining structure-preserving tree operations with reversible-jump Markov chain Monte Carlo sampling, it infers the joint posterior distribution while supporting both one-step-ahead prediction and simulation training objectives. Furthermore, physical equations can be embedded to construct gray-box models. Benchmark evaluations demonstrate that the proposed method effectively achieves dynamic system equation discovery and uncertainty quantification. Notably, the gray-box model incorporating the Morison equation significantly outperforms purely physics-driven baselines, substantially enhancing the robustness of equation discovery.

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.

Problem

• Equation discovery
• Tree-Adjoining Grammars
• Nonlinear System Identification
• Bayesian inference
• Uncertainty quantification

Innovation

• Bayesian Tree-Adjoining Grammars
• Equation Discovery
• Reversible-Jump MCMC
• Likelihood-Free Inference
• Grey-Box Modeling