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.

翻译：数字孪生作为实现复杂信息物理系统实时监测与预测性维护的核心使能技术，对其虚拟模型提出了关键要求：高预测精度、强可解释性以及在线自适应能力。然而，现有技术难以同时满足这些需求：贝叶斯方法擅长不确定性量化但缺乏模型可解释性，而可解释的符号辨识方法（例如SINDy）受限于其离线、批处理的特性，使得实时更新具有挑战性。为弥合这一语义与计算鸿沟，本文提出了一种新颖的基于贝叶斯回归的符号学习框架。该框架将在线符号发现构建为一个统一的概率状态空间模型。通过引入稀疏马蹄铁先验，模型选择被转化为贝叶斯推断任务，从而能够同时进行系统辨识与不确定性量化。此外，我们推导了一种带有遗忘因子的在线递归算法，并建立了保证后验分布适定性的精确递归条件。这些条件也作为数据效用的实时监测器，增强了算法的鲁棒性。同时，本文提供了严格的收敛性分析，证明了参数估计在持续激励条件下的收敛性。案例研究验证了所提框架在实现可解释的概率预测与在线学习方面的有效性。