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Neural Laplace: Learning diverse classes of differential equations in the Laplace domain
Samuel Holt · Zhaozhi Qian · Mihaela van der Schaar

Tue Jul 19 03:30 PM -- 05:30 PM (PDT) @ Hall E #407

Neural Ordinary Differential Equations model dynamical systems with ODEs learned by neural networks.However, ODEs are fundamentally inadequate to model systems with long-range dependencies or discontinuities, which are common in engineering and biological systems. Broader classes of differential equations (DE) have been proposed as remedies, including delay differential equations and integro-differential equations.Furthermore, Neural ODE suffers from numerical instability when modelling stiff ODEs and ODEs with piecewise forcing functions.In this work, we propose Neural Laplace, a unifying framework for learning diverse classes of DEs including all the aforementioned ones.Instead of modelling the dynamics in the time domain, we model it in the Laplace domain, where the history-dependencies and discontinuities in time can be represented as summations of complex exponentials. To make learning more efficient, we use the geometrical stereographic map of a Riemann sphere to induce more smoothness in the Laplace domain.In the experiments, Neural Laplace shows superior performance in modelling and extrapolating the trajectories of diverse classes of DEs, including the ones with complex history dependency and abrupt changes.

Author Information

Samuel Holt (University of Cambridge)
Samuel Holt

## PhD candidate in time series machine learning methods 2021 - 2025 ### University of Cambridge #### Supervisor Prof. Mihaela van der Schaar Developed and published a novel new state of the art dynamical systems time series model at ICML 2022 (selected for a long oral presentation) - Neural Laplace : [https://icml.cc/virtual/2022/oral/16728](https://icml.cc/virtual/2022/oral/16728). [Code](https://github.com/samholt/NeuralLaplace). Submitted a novel framework to scale symbolic regression to 12 + dimensions for NeurIps 2022 (under review). Continuing work on state of the art dynamical system time series models in relation to incorporation of known inductive biases to help forecasting and continuous optimal control settings for submission to ICLR 2023. **Focus areas:** Learning differential equations with sequential models, Neural Networks in the Laplace domain, Symbolic regression and model based reinforcement learning in continuous control tasks. **Supervisor:** [Prof. Mihaela van der Schaar](https://www.vanderschaar-lab.com/prof-mihaela-van-der-schaar/) **Contact**: - WhatsApp: +44 7775421326 - Email : sih31@cam.ac.uk - GitHub: https://github.com/samholt - Linkedin: https://www.linkedin.com/in/samuel-holt/ - Google Scholar: https://scholar.google.com/citations?user=Ey5aInIAAAAJ&hl=en

Zhaozhi Qian (University of Cambridge)
Mihaela van der Schaar (University of Cambridge and UCLA)

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