Modelling Dynamical Systems: Learning ODEs with No Internal ODE Resolution

The quest for accurate modelling and simulation of dynamical systems is the Holy Grail of computational physics and numerical engineering. In deep learning, main approaches proposed in the literature include prediction by time series and modelling by Ordinary Differential Equations (ODEs). The usual methods for learning optimal parameters then consist of formulating the question as a reachability problem and then optimizing some suitable cost function for this 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 dynamic, which is highly versatile and possibly discontinuous. In order to extend the scope of dynamical system modelling to meet the particle accelerator modelling challenge, we present a new approach that can cover this context called implicit neural ODE (INode). It uses the modelling of discontinuous behaviour through integral operators; these operators are used to pre-process the data to get a more 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 convergence properties under moderate assumptions.