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Advances in differentiable numerical integrators have enabled the use ofgradient descent techniques to learn ordinary differential equations (ODEs). Inthe context of machine learning, differentiable solvers are central for NeuralODEs (NODEs), a class of deep learning models with continuous depth, ratherthan discrete layers. However, these integrators can be unsatisfactorily slowand inaccurate when learning systems of ODEs from long sequences, or whensolutions of the system vary at widely different timescales in each dimension.In this paper we propose an alternative approach to learning ODEs from data: werepresent the underlying ODE as a vector field that is related to another basevector field by a differentiable bijection, modelled by an invertible neuralnetwork. By restricting the base ODE to be amenable to integration, we candrastically speed up and improve the robustness of integration. We demonstratethe efficacy of our method in training and evaluating continuous neuralnetworks models, as well as in learning benchmark ODE systems. We observeimprovements of up to two orders of magnitude when integrating learned ODEswith GPUs computation.

Learning ODEs via Diffeomorphisms for Fast and Robust Integration