Is it possible to simulate SDDE with modeling toolkit?

Specific Domains Modelling & Simulations
1

Hi,
I’ve come across Chris Rackauckas post that it is possible to simulate DDEs using Modeling Toolkit (ModelingToolkit: (Transport) delay? Example: Inflow to one water reservoir is a delayed outflow from another - #7 by filip.cerny).

Then I tried to simulate SDDE, and it worked, but the solution looked different from the solution when directly using the DifferentialEquations module. Have I made some mistake or is SDDEProblem not yet compatible with ModelingToolkit?
Thank you! I have attached the simplified code below.

Simulating with MTK:

using DifferentialEquations
using StochasticDelayDiffEq
using ModelingToolkit
using ModelingToolkit: t_nounits as t, D_nounits as D
using Random 
using Plots

Random.seed!(1)

@parameters a b τ
@variables x₁(t) x₂(..)
@brownian w₁ 

eqs = [D(x₁) ~ a * x₁ - x₁ * x₂(t-τ) + 0.2*w₁, 
       D(x₂(t)) ~ b * x₂(t) + x₁ * x₂(t)] 

# Parameters and setup
params = [a => 1., b => -3.0, τ => 0.01]
inits = [x₁ => 1.0, x₂(t) => 1.0]
tspan = (0.0, 20.0)

sys = structural_simplify(System(eqs, t; name=Symbol("LotkaVolterra")))
prob = SDDEProblem(sys, inits, tspan, params; constant_lags=[τ]) 
sol = solve(prob, SRIW1())
plot(sol, xlims=(0, 20), ylims=(0, 10))

Output:

Screenshot_3
Screenshot_31103×726 42.5 KB

Simulating with differentialEquations:

using DifferentialEquations
using Plots
using StochasticDelayDiffEq
using StochasticDiffEq
using Random 

Random.seed!(1)

function eq!(du, u, h, p, t)
    a, b, τ = p
    hu2 = h(p, t - τ)[2]
    du[1] = a * u[1] - u[1] * hu2
    du[2] = b * u[2] + u[1] * u[2] # + h(p, t - τ)[1] # -3 * Vex + Ve * Vex
end

function noise!(du, u, h, p, t)
    du1 = 0.2
    du2 = 0.
    du.= [du1, du2]
end

# Parameters and setup
a, b, τ = 1.0, -3.0, 0.01
params = [a, b, τ]
inits = [1.0, 1.0]
tspan = (0.0, 20.0)
h(p, t) = inits

prob = SDDEProblem(eq!, noise!, inits, h, tspan, params; constant_lags=τ)
sol = solve(prob, SRIW1())
plot(sol, xlims=(0, 20), ylims=(0, 10))

Output:

Screenshot_1
Screenshot_11104×727 58.9 KB

2

@cryptic.ax can you double check the codegen here?

3

Codegen all looks fine. I’ll try and dig further