Modelling Dynamical Systems Using Neural Ordinary Differential Equations

Abstract

Modelling of dynamical systems is an important problem in many fields of science. In this thesis we explore a data-driven approach to learn dynamical systems from data governed by ordinary differential equations using Neural Ordinary Differential Equations (ODENet). ODENet is a recently introduced family of artificial neural network architectures that parameterize the derivative of the input data with a neural network block. The output of the full architecture is computed using any numerical differential equation solver. We evaluate the modelling capabilities of ODENet on four datasets synthesized from dynamical systems governed by ordinary differential equations. We extract a closed-form expression for the derivative parameterized by ODENet with two different methods: a least squares regression approach and linear genetic programming. To evaluate ODENet the derivatives learned by the network were compared to the true ordinary differential equations used to synthesize the data. We found that ODENet learns a parameterization of the underlying ordinary differential equation governing the data that is valid in a region surrounding the training data. From this region a closed-form expression that was close to the true system could be extracted for both linear and non-linear ODEs.