See the writings of the indefatigable Chris Rackauckas (Rackauckas et al. 2018), plus his tutorial implementations and whole MIT course. Chris Rackauckas’ lecture notes christen this area “scientific machine learning,” which is a confusing and ambiguous choice, but let us run with it.

3 Basic SDEs

TBD.

4 In PDEs

Learning stochastic partial differential equations where a whole random field evolves in time is something of interest to me; see spatiotemporal nets and spatiotemporal dynamics for more on that theme.

See differentiable PDE solvers for now.

5 With sparse SDEs

For least-squares system identification see sparse stochastic processes.

6 Neural approximation

See neural learning dynamics.

7 Via simulations only

See simulation-based inference.

8 Controlled differential equations

TBD

9 Method of adjoints

A trick in differentiation which happens to be useful in differentiating likelihood (or other functions) of time-evolving systems e.g. Errico (1997). If the system is deterministic, this is not too bad. It is complicated in stochastic systems.

For now, see the method of adjoints in the autodiff notebook.

10 Tooling

10.1 Python

Diffrax

Diffrax is a JAX-based library providing numerical differential equation solvers.

Features include:

ODE/SDE/CDE (ordinary/stochastic/controlled) solvers

lots of different solvers (including Tsit5, Dopri8, symplectic solvers, implicit solvers)

vmappable everything (including the region of integration)

using a PyTree as the state

dense solutions

multiple adjoint methods for backpropagation

support for neural differential equations.

From a technical point of view, the internal structure of the library is pretty cool — all kinds of equations (ODEs, SDEs, CDEs) are solved in a unified way (rather than being treated separately), producing a small tightly-written library.

torchdyn (docs).

10.2 Julia

Chris Rackauckas is a veritable wizard with this stuff; read his blog.

Here is a tour of fun tricks with stochastic PDEs. There is a lot of tooling for this; DiffEqOperators … does something. DiffEqFlux (EZ neural ODEs works with Flux and claims to make neural SDE simple.

+1 for Julia here.

11 Incoming

Bayesian Model of Planetary Motion: exploring ideas for a modeling workflow when dealing with ordinary differential equations and multimodality

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