Support for continuous spatial models

[Homo-carbonis]

Hello, I am working on a package ReactionDiffusion.jl, which uses Catalyst as the front end to a custom-made solver for 1D reaction-diffusion systems. Catalyst makes describing things like gene regulatory networks really easy, so it's ideal for our purposes.

At the moment we're using Catalyst to specify reaction kinetics and generate an ODE, and our own macros for the spatial mechanics, but it would be great if we could use the Catalyst DSL for everything. I was wondering if there are plans to better support modelling spatial systems on continuous domains? I would be quite interested in contributing if so.

I experimented with using the existing LatticeReactionSystem objects but there are a few things which are either not yet available, or awkward from the point of view of spatially continuous systems:

1. Discretisation not baked into the model, so you can pass num_verts to the solver.
2. Parameters for the size of the domain.
3. Neumann boundary conditions.
4. Continuously varying initial conditions.

I think maybe the ideal situation would be (having defined @species U(x,t)) to be able to write expressions for U(x,0), Uₓ(0,t), Uₓ(L,t), and the diffusion coefficients within the main DSL.

[TorkelE]

Hello, quick reply, and can come back with more later on.

The current LatticeReactionSystem interface is currently primarily there as "something" which can handle spatial systems, and especially considering discrete spaces (e.g. transportation between well-mixed containers). The idea is that it is something that is powerful enough to allow users to play around with and use Catalyst in a spatial context. It is also a relatively nice interface to the spatial jump solvers in JumpProcesses.jl

It is very much not intended to be a "final" interface for spatial modelling, more something to have until we finally can get there.

So yes, how a (more) "final" form of spatial modelling in Catalyst should look is very much an open discussion and any form of input and engagement is welcome - this is definitely something ti would be useful to discuss closer.

[Homo-carbonis]

Thank you, that's good to know. I've written some simple @diffusion_system and @initial_conditions macros for specifying these things in a discretisation-independent way, but it seems a bit redundant to develop them further when Catalyst is already working towards supporting such things.

[TorkelE]

One thing that has vaguely been holding us back is also that we think that MTK have intended to have more general PDE support. If/when this gets implemented, we could simply lower into that, which might simplify things. However, I am not sure what the plans and timelines are here.

[isaacsas]

We shouldn't wait on MTK for this -- I'm not aware that anyone is really actively working on it anymore (beyond NeuralPDEs type stuff). It would be better to get a nice Catalyst representation for spatial systems that is independent of lowering, and then setup custom lowering for different PDE solvers as best we can.

[ChrisRackauckas]

Is there anything required here that isn't handled by MOL already?

[Homo-carbonis]

Not as far as I'm aware, but I have only tried it for 1D systems so far.

Our package uses a pseudo-spectral method, so I've only looked at MOL to check we're getting the same results.