Gridap.jl VS NeuralPDE.jl

Another I've been working on learning is Julia, which aims to use a syntax very similar to how you'd write it mathematically, and I like being able to include units in calculations using the unitful.jl package, and there are FEM packages available like Gridap.

PINN methods are absurdly slow (DeepXDE is about 10,000x slower than an ODE solver for example, while using implicit parallelism vs serial ODE solver) but they are flexible. So ModelingToolkit.jl has alternative options, like DiffEqOperators.jl takes the same specification and generates ODESystem and NonlinearSystem problems via finite difference discretizations (known as "method of lines"). There's a (pseudo-)spectral part of the interface coming relatively soon as well, with GridAP.jl integration for FEM coming soon. So this is made to be a universal arbitrary PDE -> approximate solution interface which is generic to the method and solving process.

The documentation has a manifest associated with it: https://docs.sciml.ai/NeuralPDE/dev/#Reproducibility. Instantiating the manifest will give you all of the exact versions used for the documentation build (https://github.com/SciML/NeuralPDE.jl/blob/gh-pages/v5.7.0/assets/Manifest.toml). You just ]instantiate folder_of_manifest. Or you can use the Project.toml.

PDE solving libraries are MethodOfLines.jl and NeuralPDE.jl. NeuralPDE is very general but not very fast (it's a limitation of the method, PINNs are just slow). MethodOfLines is still somewhat under development but generates quite fast code.

GitHub - SciML/NeuralPDE.jl: Physics-Informed Neural Networks (PINN) and Deep BSDE Solvers of Differential Equations for Scientific Machine Learning (SciML) accelerated simulation

That doesn't mean there's no use case for PINNs, we wrote a giant review-ish kind of thing on NeuralPDE.jl to describe where PINNs might be useful. It's just... not the best for publishing. It's things like, (a) where you have not already optimized a classical method, (b) need something that's easy to generate solvers for different cases without too much worry about stability, (c) high dimensional PDEs, and (d) surrogates over parameters. (c) and (d) are the two "real" uses cases you can actually publish about, but they aren't quite good for (c) (see mesh-free methods from the old radial basis function literature in comparison) or (d) (there are much faster surrogate techniques). So we are continuing to work on them for (a) and (b) as an interesting option as part of a software suite, but that's not the kind of thing that's really publishable so I don't think we plan to ever submit that article anywhere.

Did this change at all with the advent of Physics Informed Neural Networks? The Julia language has some really impressive tools for that use case. https://github.com/SciML/NeuralPDE.jl

PDEs are difficult because you don't have a simple numerical definition over all PDEs because they can be defined by arbitrarily many functions. u' = Laplace u + f? Define f. u' = g(u) * Laplace u + f? Define f and g. Etc. To cover the space of PDEs you have to go symbolic at some point, and make the discretization methods dependent on the symbolic form. This is precisely what the ModelingToolkit.jl ecosystem is doing. One instantiation of a discretizer on this symbolic form is NeuralPDE.jl which takes a symbolic PDESystem and generates an OptimizationProblem for a neural network which represents the solution via a Physics-Informed Neural Network (PINN).

NeuralPDE.jl fully automates the approach (and extensions of it, which are required to make it solve practical problems) from symbolic descriptions of PDEs, so that might be a good starting point to both learn the practical applications and get something running in a few minutes. As part of MIT 18.337 Parallel Computing and Scientific Machine Learning I gave an early lecture on physics-informed neural networks (with a two part video) describing the approach, how it works and what its challenges are. You might find those resources enlightening.

What is great about ModelingToolkit.jl is how its used in practical ways for other packages. E.g. NeuralPDE.jl.

Compared to SymPy, I feel that it is less of a "how do I integrate this function" package and more about "how can I build this DSL" framework.

https://github.com/SciML/NeuralPDE.jl