Scientific machine learning for inferring dynamical systems combines data-driven modeling, physics-based modeling, and empirical knowledge. It plays an essential role in engineering design and digital twinning. In this work, we primarily focus on an operator inference methodology that builds dynamical models, preferably in low-dimension, with a prior hypothesis on the model structure, often determined by known physics or given by experts. Then, for inference, we aim to learn the operators of a model by setting up an appropriate optimization problem. One of the critical properties of dynamical systems is stability. However, this property is not guaranteed by the inferred models. In this work, we propose inference formulations to learn quadratic models, which are stable by design. Precisely, we discuss the parameterization of quadratic systems that are locally and globally stable. Moreover, for quadratic systems with no stable point yet bounded (e.g., chaotic Lorenz model), we discuss how to parameterize such bounded behaviors in the learning process. Using those parameterizations, we set up inference problems, which are then solved using a gradient-based optimization method. Furthermore, to avoid numerical derivatives and still learn continuous systems, we make use of an integral form of differential equations. We present several numerical examples, illustrating the preservation of stability and discussing its comparison with the existing state-of-the-art approach to infer operators. By means of numerical examples, we also demonstrate how the proposed methods are employed to discover governing equations and energy-preserving models.

翻译：用于推断动力系统的科学机器学习融合了数据驱动建模、基于物理的建模和经验知识。该方法在工程设计及数字孪生中具有核心作用。本研究主要聚焦于算子推断方法学——该方法基于模型结构的先验假设（通常由已知物理规律或专家经验确定），构建低维动力模型。在推断过程中，我们通过构建合适的优化问题来学习模型算子。动力系统的一个关键特性是稳定性，然而推断模型无法保证此特性。本文提出可保证设计稳定性的二次模型推断框架。具体而言，我们讨论了局部稳定与全局稳定的二次系统的参数化方法。此外，针对无稳定点但有界二次系统（如混沌Lorenz模型），我们阐述了如何在学习过程中参数化此类有界行为。利用这些参数化方法，我们构建了推断问题，并采用基于梯度的优化方法进行求解。为避免数值导数同时保持连续系统学习能力，我们采用微分方程的积分形式。通过多个数值算例，我们展示了稳定性的保持效果，并与现有最优算子推断方法进行对比分析。数值算例还进一步证明了所提方法在发现控制方程和能量守恒模型中的应用价值。