Learning stochastic differential equations. Related: Analysing a neural net itself as a dynamical system, which is not quite the same but crosses over. Variational state filters.
A deterministic version of this problem is what e.g. the famous Vector Institute Neural ODE paper (Chen et al. 2018) did. Author Duvenaud argues that in some ways the hype ran away with the Neural ODE paper, and credits CasADI with 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.
Also, we would prefer models to be stochastic. Learnable SDEs are probably what we want. I’m particularly interested on jump ODE regression.
Homework: Duvenaud again, tweeting some explanatory animations.
Turtorials on this are available. (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.
Note connection to reparameterization tricks, in that neural ODEs give you cheap differentiable reparameterizations.
Questions
How do you do ensemble training for posterior predictives in NODEs? How do you guarantee stability in the learned dynamics?
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