In many problems of data-driven modeling for dynamical systems, the governing equations are not known a priori and must be selected phenomenologically from a large set of candidate interactions and basis functions. In such situations, point estimates alone can be misleading, because multiple model components may explain the observed data comparably well, especially when the data are limited or the dynamics exhibit poor identifiability. Quantifying the uncertainty associated with model selection is therefore essential for constructing reliable dynamical models from data. In this work, we develop a Bayesian sparse identification framework for dynamical systems with coupled components, aimed at inferring both interaction structure and functional form together with principled uncertainty quantification. The proposed method combines sparse modeling with Bayesian model averaging, yielding posterior inclusion probabilities that quantify the credibility of each candidate interaction and basis component. Through numerical experiments on oscillator networks, we show that the framework accurately recovers sparse interaction structures with quantified uncertainty, including higher-order harmonic components, phase-lag effects, and multi-body interactions. We also demonstrate that, even in a phenomenological setting where the true governing equations are not contained in the assumed model class, the method can identify effective functional components with quantified uncertainty. These results highlight the importance of Bayesian uncertainty quantification in data-driven discovery of dynamical models.

翻译：在动力学系统数据驱动建模的诸多问题中，控制方程并非先验已知，而需要从大量候选相互作用和基函数中唯象地进行选择。在此类情形下，仅依赖点估计可能导致误判，因为多个模型组件可能对观测数据具有同等解释力——尤其当数据有限或动力学系统可辨识性较差时。因此，量化模型选择过程中的不确定性，对于构建可靠的数据驱动动力学模型至关重要。本研究针对具有耦合组件的动力学系统，提出了一种贝叶斯稀疏辨识框架，旨在同时推断相互作用结构与函数形式，并进行原则性不确定性量化。该方法将稀疏建模与贝叶斯模型平均相结合，通过后验包含概率量化每个候选相互作用和基分量的可信度。基于振荡器网络的数值实验表明，该框架能够准确恢复具有量化不确定性的稀疏相互作用结构，包括高阶谐波分量、相位滞后效应及多体相互作用。此外，即使在被假设模型类不包含真实控制方程的唯象建模场景中，该方法仍能识别具有量化不确定性的有效函数组件。这些结果凸显了贝叶斯不确定性量化在数据驱动动力学模型发现中的关键作用。