Physics-based and first-principles models pervade the engineering and physical sciences, allowing for the ability to model the dynamics of complex systems with a prescribed accuracy. The approximations used in deriving governing equations often result in discrepancies between the model and sensor-based measurements of the system, revealing the approximate nature of the equations and/or the signal-to-noise ratio of the sensor itself. In modern dynamical systems, such discrepancies between model and measurement can lead to poor quantification, often undermining the ability to produce accurate and precise control algorithms. We introduce a discrepancy modeling framework to identify the missing physics and resolve the model-measurement mismatch with two distinct approaches: (i) by learning a model for the evolution of systematic state-space residual, and (ii) by discovering a model for the deterministic dynamical error. Regardless of approach, a common suite of data-driven model discovery methods can be used. The choice of method depends on one's intent (e.g., mechanistic interpretability) for discrepancy modeling, sensor measurement characteristics (e.g., quantity, quality, resolution), and constraints imposed by practical applications (e.g., modeling approaches using the suite of data-driven modeling methods on three continuous dynamical systems under varying signal-to-noise ratios. Finally, we emphasize structural shortcomings of each discrepancy modeling approach depending on error type. In summary, if the true dynamics are unknown (i.e., an imperfect model), one should learn a discrepancy model of the missing physics in the dynamical space. Yet, if the true dynamics are known yet model-measurement mismatch still exists, one should learn a discrepancy model in the state space.

翻译：基于物理机制和第一性原理的模型广泛存在于工程与物理科学领域，能够以指定精度对复杂系统动力学进行建模。在推导控制方程时采用的近似方法，常导致模型与基于传感器的系统测量值之间存在差异，这揭示了方程本身的近似特性及/或传感器自身的信噪比局限性。在现代动力系统中，此类模型与测量值之间的偏差可能导致量化效果不佳，往往削弱生成精准控制算法的能力。本文提出一种差异建模框架，通过两种不同路径识别缺失物理机制并解决模型-测量失配问题：(i) 学习系统状态空间残差的演化模型；(ii) 发现确定性动力学误差的模型。无论采用何种路径，均可使用通用的数据驱动模型发现方法。具体方法的选择取决于差异建模的目标（如机制可解释性）、传感器测量特性（如数量、质量、分辨率）以及实际应用约束（如建模方法）。我们在三种不同信噪比条件下的连续动力系统中，采用多种数据驱动建模方法验证了该框架的有效性。最后，我们根据不同误差类型指出了每种差异建模方法的结构性缺陷。总结而言：若真实动力学未知（即模型不完善），应在动力学空间中学习缺失物理机制的差异模型；若真实动力学已知但模型-测量失配仍存在，则应在状态空间中学习差异模型。