Modelling Dynamical Systems: Learning ODEs with No Internal ODE Resolution

EasyChair Preprint 15198

Abstract

The quest for accurate modeling and simulation of dynamical systems is the Holy Grail of computational physics and numerical engineering. In deep learning, the two main approaches proposed in the literature include time series prediction and ODE modelling. Methods are based on learning (possibly locally) optimal parameters that minimize the cost associated with some suitable reachability problem. However, these two approaches fail to model specific complex dynamical systems. The presented work considers the case of modelling and predicting the behaviour of beams in particle accelerators. The difficulty lies in the associated dynamics’ highly versatile and possibly discontinuous behaviours. Extending the scope of dynamical system modelling to meet the particle accelerator modelling challenge, we present a new approach called Implicit Neural ODE (INode). INode approaches the modelling of discontinuous behaviour through integral operators; these operators are used to pre-process the data and define a classical regression problem. Finally, the global model of the dynamical system is formulated as the solution of an ODE, which contains the solution of the regression problem. INode thus enables the learning of a data-driven ODE while removing the computationally heavy ODE resolution from the training loop. The formal analysis of the approach establishes its consistency and con- vergence properties under moderate assumptions.

Keyphrases: Universal approximation theorem, continous deep learning model, learning, ordinary differential equation

@booklet{EasyChair:15198,
  author    = {Johanne Cohen and Emmanuel Goutierre and Hayg Guler and Fotios Kapotos and Sida-Bastien Li and Michèle Sébag and Bowen Zhu},
  title     = {Modelling Dynamical Systems: Learning ODEs with No Internal ODE Resolution},
  howpublished = {EasyChair Preprint 15198},
  year      = {EasyChair, 2024}}