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Neural Ordinary Differential Equations model dynamical systems with ODEslearned by neural networks. However, ODEs are fundamentally inadequate to modelsystems with long-range dependencies or discontinuities, which are common inengineering and biological systems. Broader classes of differential equations(DE) have been proposed as remedies, including delay differential equations andintegro-differential equations. Furthermore, Neural ODE suffers from numericalinstability when modelling stiff ODEs and ODEs with piecewise forcingfunctions. In this work, we propose Neural Laplace, a unified framework forlearning diverse classes of DEs including all the aforementioned ones. Insteadof modelling the dynamics in the time domain, we model it in the Laplacedomain, where the history-dependencies and discontinuities in time can berepresented as summations of complex exponentials. To make learning moreefficient, we use the geometrical stereographic map of a Riemann sphere toinduce more smoothness in the Laplace domain. In the experiments, NeuralLaplace shows superior performance in modelling and extrapolating thetrajectories of diverse classes of DEs, including the ones with complex historydependency and abrupt changes.

Neural Laplace: Learning diverse classes of differential equations in the Laplace domain