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

Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inference