DataDrivenDiffEq.jl VS Catalyst.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.

ModelingToolkit.jl adds a different spin on this by noting what makes a good modeling system isn't top down but a system that allows for bottom up contributions. ModelingToolkit is built on Symbolics.jl which uses OSCAR.jl etc., so every time the symbolics community gets better ModelingToolkit.jl gets better. It connects to the whole SciML ecosystem, so any improvement to any of the SciML interface packages is directly an improvement to ModelingToolkit.jl. ModelingToolkit is made to be a set of composable compiler abstractions called transformations, so anyone can add new packages that do new transformations that improve the ecosystem. One that I really like is MomentClosure.jl which symbolically transforms stochastic ModelingToolkit models (ReactionSystem) to approximate symbolic ODESystem models of the moments. And there's domain-specific langauges like Catalyst.jl being built on the interface to give more ways to build models, which is spawning the biocommunity to make model importers into the symbolic forms, when then feeds more ODE models into the same compiler. JuliaSim is then building on this ecosystem, adding cloud infrastructure that is special-purpose made for doing parallel computations of these models, automatic symbolic model discovery from data, automatic generation of approximate models with machine learning, and tying the Julia Computing compiler team into the web that is building this ecosystem.