Online identification of nonlinear time-varying systems with uncertain information

2026-01-15

AI Summary

Existing digital twin approaches struggle to simultaneously achieve high predictive accuracy, strong interpretability, and online adaptability. This work proposes a Bayesian Regression Symbolic Learning (BRSL) framework that formulates online symbolic discovery as a probabilistic state-space model. By incorporating a sparse horseshoe prior, the method recasts model selection as Bayesian inference, enabling concurrent system identification and uncertainty quantification. The approach innovatively integrates Bayesian sparse inference with online symbolic regression and introduces a recursive algorithm with a forgetting factor to support real-time updates. Furthermore, it establishes recursive conditions for posterior well-posedness as a mechanism to monitor data utility and provides theoretical guarantees on the convergence of parameter estimates under persistent excitation. Experimental results demonstrate that the proposed method delivers accurate, interpretable, and probabilistically robust predictions with efficient online learning capabilities across multiple case studies.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Digital twins (DTs), serving as the core enablers for real-time monitoring and predictive maintenance of complex cyber-physical systems, impose critical requirements on their virtual models: high predictive accuracy, strong interpretability, and online adaptive capability. However, existing techniques struggle to meet these demands simultaneously: Bayesian methods excel in uncertainty quantification but lack model interpretability, while interpretable symbolic identification methods (e.g., SINDy) are constrained by their offline, batch-processing nature, which make real-time updates challenging. To bridge this semantic and computational gap, this paper proposes a novel Bayesian Regression-based Symbolic Learning (BRSL) framework. The framework formulates online symbolic discovery as a unified probabilistic state-space model. By incorporating sparse horseshoe priors, model selection is transformed into a Bayesian inference task, enabling simultaneous system identification and uncertainty quantification. Furthermore, we derive an online recursive algorithm with a forgetting factor and establish precise recursive conditions that guarantee the well-posedness of the posterior distribution. These conditions also function as real-time monitors for data utility, enhancing algorithmic robustness. Additionally, a rigorous convergence analysis is provided, demonstrating the convergence of parameter estimates under persistent excitation conditions. Case studies validate the effectiveness of the proposed framework in achieving interpretable, probabilistic prediction and online learning.

Problem

• online identification
• nonlinear time-varying systems
• uncertainty quantification
• model interpretability
• symbolic learning

Innovation

• Bayesian symbolic regression
• online system identification
• digital twins
• horseshoe prior
• recursive state-space model