Generalize Deep Splitting Algorithm

[ChrisRackauckas]

https://github.com/vboussange/HighDimPDE.jl/blob/main/src/DeepSplitting.jl#L110

This is hardcoded to Euler-Maruyama. This could potentially be improved by making this use StochasticDiffEq to generalize the time stepping algorithms which could be used. It would be interesting research wise to just see whether a higher order method with a larger dt ends up being more efficient here.

[ChrisRackauckas]

Could multilevel Picard be generalized as well?

https://github.com/vboussange/HighDimPDE.jl/blob/main/src/MLP.jl#L263

I don't know that one as much.

[vboussange]

Both are interesting ideas. MLP could potentially be generalised, although I have some concerns about the numerical cost, as it heavily relies on nested calls.

[ChrisRackauckas-Claude]

Status Update on Issue #5

cc @ChrisRackauckas

I investigated this issue and wanted to provide a status update:

Current State

The Euler-Maruyama time-stepping is still hardcoded in two locations:

• src/DeepSplitting.jl:160: y1 .= y0 .+ μ(y0, p, t) .* dt .+ σ(y0, p, t) .* sqrt(dt) .* dW
• src/MLP.jl:356: x2 .= x + (prob.μ(x, prob.p, t) .* dt .+ prob.σ(x, prob.p, t) .* sqrt(dt) .* x2)

Existing Work

• PR update sde_loop by removing the loop and use EnsembleProblem: https://github.com/SciML/HighDimPDE.jl/issues/5 #93 attempted to address this issue (April 2024) but:

  • Currently has merge conflicts
  • Received feedback that the code wasn't properly written as functions and may not have been tested
  • Has been stale for ~8 months

• StochasticDiffEq.jl is already a dependency, so integration is feasible

Questions

1. Is anyone actively working on this feature?
2. Should PR update sde_loop by removing the loop and use EnsembleProblem: https://github.com/SciML/HighDimPDE.jl/issues/5 #93 be revisited/rebased, or would a fresh implementation be preferred?
3. For the MLP algorithm (mentioned by @ChrisRackauckas in an earlier comment), should this generalization be included in the same effort, or handled separately given concerns about "numerical cost due to nested calls"?

This would be an interesting research direction as mentioned in the original issue - comparing whether higher-order methods with larger dt could be more efficient for these high-dimensional PDE solvers.