DataDrivenDiffEq.jl VS DifferentialEquations.jl

If you are looking to infer the actual structure (not just parameters) of an ODE given some data, there is DataDrivenDiffEq.jl. https://github.com/SciML/DataDrivenDiffEq.jl

This is all not to mention the fact that PINNs are a notoriously computationally intensive approach, where it's pretty easy to show the differentiable solver approach of DiffEqFlux.jl achieves about a 10,000x speedup over another PINN package on parameter estimation of Lorenz equations, and while it scales to higher PDE dimensions well, it doesn't scale to larger systems of PDEs very well. You'll want to factor in a good chunk of training time, and of course increase that by a few orders of magnitude if your dynamics are stiff. Altogether, without knowing your exact problem it's hard to give a rough idea of how practical it would be, but if I tasked a beginning graduate student with trying this out on some of the biological PDEs I work with, then I would give them about 4-6 months to get something decent together.

And for reference implementations you can take a look at DataDrivenDiffEq.jl. All DMDs (that I know of) essentially work by building and solving a convex optimization.

This lists some of its unique abilities:

https://docs.sciml.ai/DiffEqDocs/stable/

The routines are sufficiently generic, with regard to Julia’s type system, to allow the solvers to automatically compose with other packages and to seamlessly use types other than Numbers. For example, instead of handling just functions Number→Number, you can define your ODE in terms of quantities with physical dimensions, uncertainties, quaternions, etc., and it will just work (for example, propagating uncertainties correctly to the solution¹). Recent developments involve research into the automated selection of solution routines based on the properties of the ODE, something that seems really next-level to me.

[1] https://lwn.net/Articles/834571/

https://github.com/SciML/DifferentialEquations.jl/issues/786. As you could see from the tweet, it's now at 0.1 seconds. That has been within one year.

Also, if you take a look at a tutorial, say the tutorial video from 2018,

It's not faith, and it's not all from Julia itself. https://github.com/SciML/DifferentialEquations.jl/issues/785 should reduce compile times of what OP mentioned for example.

Let's even put raw numbers to it. DifferentialEquations.jl usage has seen compile times drop from 22 seconds to 3 seconds over the last few months.

https://github.com/SciML/DifferentialEquations.jl/issues/786

https://github.com/SciML/DifferentialEquations.jl/issues/737 double posted, with the answer here. Please don't do that.