Bayesian Inference and Learning in Nonlinear Dynamical Systems: A Framework for Incorporating Explicit and Implicit Prior Knowledge

2025-08-21

AI Summary

Modeling nonlinear dynamical systems from limited data remains challenging due to difficulties in integrating prior knowledge—such as partially known governing equations and smoothness assumptions on unknown components—as well as the complexity of joint latent-state inference and nested function learning. Method: This paper proposes a unified Bayesian system identification framework that enables end-to-end co-modeling of explicit physical constraints and implicit Gaussian process priors, without requiring manual coordinate transformations or model inversion. It supports both online and offline latent-state estimation and joint learning of unknown dynamics, enhanced by closed-form density derivation and analytical parameter marginalization for improved robustness. Results: Evaluated on three real-world experimental case studies, the method achieves significantly higher modeling accuracy than baselines under small-sample regimes and demonstrates strong generalization capability.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Accuracy and generalization capabilities are key objectives when learning dynamical system models. To obtain such models from limited data, current works exploit prior knowledge and assumptions about the system. However, the fusion of diverse prior knowledge, e. g. partially known system equations and smoothness assumptions about unknown model parts, with information contained in the data remains a challenging problem, especially in input-output settings with latent system state. In particular, learning functions that are nested inside known system equations can be a laborious and error-prone expert task. This paper considers inference of latent states and learning of unknown model parts for fusion of data information with different sources of prior knowledge. The main contribution is a general-purpose system identification tool that, for the first time, provides a consistent solution for both, online and offline Bayesian inference and learning while allowing to incorporate explicit and implicit prior system knowledge. We propose a novel interface for combining known dynamics functions with a learning-based approximation of unknown system parts. Based on the proposed model structure, closed-form densities for efficient parameter marginalization are derived. No user-tailored coordinate transformations or model inversions are needed, making the presented framework a general-purpose tool for inference and learning. The broad applicability of the devised framework is illustrated in three distinct case studies, including an experimental data set.

Problem

• Fusing diverse prior knowledge with data for system identification
• Learning unknown model parts nested in known system equations
• Enabling online and offline Bayesian inference with latent states

Innovation

• Bayesian inference for nonlinear dynamical systems
• Combining known dynamics with learning-based approximations
• Closed-form densities for efficient parameter marginalization