Learning differntial equations, either ordinary or stochastic, using nonparametric neural techniques. This is what e.g. the famous Vector Institute Neural ODE paper (Chen et al. 2018) did, although I’m not sure its as novel as they imply, since it does look like a lot of earlier work. Related: Analysing a neural net itself as a dynamical system, whcih is not quite the same but crosses over.
Author Duvenaud argues that in some ways the hype ran away with the Neural ODE paper, and credits casadi with some of the innovations here. There are various laypersons’ introductions/ tutorials in this area, including the simple and practical magical take in julia. See also the CASADI example.
Learning an ODE in particular a purely deterministic process, feels unsatisfying; We want a model which encodes responses,and effects to interactions. It is not ideal to have time series models which need to encode everything in an initial state.
There are syntheses of these approaches that try to do everything with ODEs, all the time. (Rackauckas et al. 2018; Niu, Horesh, and Chuang 2019), and even some tutorial implementations by the indefatigable Chris Rackauckas, and a whole MIT course. Chris Rackauckas’ lecture notes christen this development “scientific machine learning”.
Learning stochastic partial differential equations where a whole random field evolves in time is something of interest to me; see spatiotemporal nets.
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