  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.
