Equation discovery with Bayesian tree-adjoining grammars

Model-based approaches for (bio)process systems often suffer from incomplete knowledge of the underlying physical, chemical, or biological laws. Universal differential equations, which embed neural networks within differential equations, have emerged as powerful tools to learn this missing physics from experimental data. However, neural networks are inherently opaque, motivating their post-processing via symbolic regression to obtain interpretable mathematical expressions. Genetic algorithm-based symbolic regression is a popular approach for this post-processing step, but provides only point estimates and cannot quantify the confidence we should place in a discovered equation. We address this limitation by applying Bayesian symbolic regression, which uses Reversible Jump Markov Chain Monte Carlo to sample from the posterior distribution over symbolic expression trees. This approach naturally quantifies uncertainty in the recovered model structure. We demonstrate the methodology on a Lotka-Volterra predator-prey system and then show how a well-designed experiment leads to lower uncertainty in a fed-batch bioreactor case study.

Our ability to predict, control, and ultimately understand complex systems rests on discovering the equations that govern their dynamics. Identifying these equations directly from noisy, limited observations has therefore become a central challenge in data-driven science, yet existing library-based sparse regression methods force a compromise between automation, statistical rigor, and computational efficiency. Here we develop Bayesian-ARGOS, a hybrid framework that reconciles these demands by combining rapid frequentist screening with focused Bayesian inference, enabling automated equation discovery with principled uncertainty quantification at a fraction of the computational cost of existing methods. Tested on seven chaotic systems under varying data scarcity and noise levels, Bayesian-ARGOS outperforms two state-of-the-art methods in most scenarios. It surpasses SINDy in data efficiency for all systems and noise tolerance for six out of the seven, with a two-order-of-magnitude reduction in computational cost compared to bootstrap-based ARGOS. The probabilistic formulation additionally enables a suite of standard statistical diagnostics, including influence analysis and multicollinearity detection that expose failure modes otherwise opaque. When integrated with representation learning (SINDy-SHRED) for high dimensional sea surface temperature reconstruction, Bayesian-ARGOS increases the yield of valid latent equations with significantly improved long horizon stability. Bayesian-ARGOS thus provides a principled, automated, and computationally efficient route from scarce and noisy observations to interpretable governing equations, offering a practical framework for equation discovery across scales, from benchmark chaotic systems to the latent dynamics underlying global climate patterns.

This paper proposes a Bayesian filtering-based approach for learning the dynamics of a physical system from partial, noisy measurements. We model the system dynamics using a Lagrangian mechanics formulation. As in Lagrangian neural networks (LNNs), we parameterize the kinetic and potential energies with neural networks. The unknown external forces in the Lagrangian formulation are modeled as white Gaussian noise. The corresponding Euler--Lagrange equations then yield a continuous-time stochastic state-space model (SSM) that describes the system dynamics. The neural network parameters and system states are then jointly learned via a maximum-likelihood method using Gaussian-approximation-based Bayesian filters. The effectiveness of the proposed method is demonstrated on pendulum and Duffing oscillator examples, and its performance is compared with conventional LNNs and with approximate Bayesian filters using known system models.