In this work, we blend machine learning and dictionary-based learning with numerical analysis tools to discover differential equations from noisy and sparsely sampled measurement data of time-dependent processes. We use the fact that given a dictionary containing large candidate nonlinear functions, dynamical models can often be described by a few appropriately chosen basis functions. As a result, we obtain parsimonious models that can be better interpreted by practitioners, and potentially generalize better beyond the sampling regime than black-box modelling. In this work, we integrate a numerical integration framework with dictionary learning that yields differential equations without requiring or approximating derivative information at any stage. Hence, it is utterly effective for corrupted and sparsely sampled data. We discuss its extension to governing equations, containing rational nonlinearities that typically appear in biological networks. Moreover, we generalized the method to governing equations subject to parameter variations and externally controlled inputs. We demonstrate the efficiency of the method to discover a number of diverse differential equations using noisy measurements, including a model describing neural dynamics, chaotic Lorenz model, Michaelis-Menten kinetics and a parameterized Hopf normal form.
Keywords: dynamical systems; machine learning; sparse regression; system identification.
© 2022 The Authors.