Parallel Ensemble Simulations

Docs: https://diffeq.sciml.ai/stable/features/ensemble/

Solving an ODE With Different Initial Conditions

Solving \(\dot{u} = 1.01u\) with \(u(0)=0.5\) and \(t \in [0, 1]\) with multiple initial conditions.

using OrdinaryDiffEq
using Plots

Linear ODE which starts at 0.5 and solves from t=0.0 to t=1.0

prob = ODEProblem((u, p, t) -> 1.01u, 0.5, (0.0, 1.0))

ODEProblem with uType Float64 and tType Float64. In-place: false
Non-trivial mass matrix: false
timespan: (0.0, 1.0)
u0: 0.5

Define a new problem for each trajectory using remake() The initial conditions (u0) are changed in this example

function prob_func(prob, i, repeat)
    remake(prob, u0=rand() * prob.u0)
end

prob_func (generic function with 1 method)

The function could also capture the necessary data outside.

initial_conditions = range(0, stop=1, length=100)

function prob_func(prob, i, repeat)
  remake(prob, u0=initial_conditions[i])
end

Define an ensemble problem

ensemble_prob = EnsembleProblem(prob; prob_func=prob_func)

EnsembleProblem with problem ODEProblem

Ensemble simulations are solved by multithreading by default

sim = solve(ensemble_prob, trajectories=100)

EnsembleSolution Solution of length 100 with uType:
SciMLBase.ODESolution{Float64, 1, Vector{Float64}, Nothing, Nothing, Vector{Float64}, Vector{Vector{Float64}}, Nothing, SciMLBase.ODEProblem{Float64, Tuple{Float64, Float64}, false, SciMLBase.NullParameters, SciMLBase.ODEFunction{false, SciMLBase.AutoSpecialize, Main.var"##231".var"#1#2", LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}, SciMLBase.StandardODEProblem}, OrdinaryDiffEqCore.CompositeAlgorithm{0, Tuple{OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}, OrdinaryDiffEqVerner.Vern7{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}, OrdinaryDiffEqRosenbrock.Rosenbrock23{0, ADTypes.AutoFiniteDiff{Val{:forward}, Val{:forward}, Val{:hcentral}, Nothing, Nothing, Int64}, Nothing, typeof(OrdinaryDiffEqCore.DEFAULT_PRECS), Val{:forward}(), true, nothing, typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!)}, OrdinaryDiffEqRosenbrock.Rodas5P{0, ADTypes.AutoFiniteDiff{Val{:forward}, Val{:forward}, Val{:hcentral}, Nothing, Nothing, Int64}, Nothing, typeof(OrdinaryDiffEqCore.DEFAULT_PRECS), Val{:forward}(), true, nothing, typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!)}, OrdinaryDiffEqBDF.FBDF{5, 0, ADTypes.AutoFiniteDiff{Val{:forward}, Val{:forward}, Val{:hcentral}, Nothing, Nothing, Int64}, Nothing, OrdinaryDiffEqNonlinearSolve.NLNewton{Rational{Int64}, Rational{Int64}, Rational{Int64}, Nothing}, typeof(OrdinaryDiffEqCore.DEFAULT_PRECS), Val{:forward}(), true, nothing, Nothing, Nothing, typeof(OrdinaryDiffEqCore.trivial_limiter!)}, OrdinaryDiffEqBDF.FBDF{5, 0, ADTypes.AutoFiniteDiff{Val{:forward}, Val{:forward}, Val{:hcentral}, Nothing, Nothing, Int64}, LinearSolve.KrylovJL{typeof(Krylov.gmres!), Int64, Nothing, Tuple{}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}, OrdinaryDiffEqNonlinearSolve.NLNewton{Rational{Int64}, Rational{Int64}, Rational{Int64}, Nothing}, typeof(OrdinaryDiffEqCore.DEFAULT_PRECS), Val{:forward}(), true, nothing, Nothing, Nothing, typeof(OrdinaryDiffEqCore.trivial_limiter!)}}, OrdinaryDiffEqCore.AutoSwitchCache{Tuple{OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}, OrdinaryDiffEqVerner.Vern7{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}}, Tuple{OrdinaryDiffEqRosenbrock.Rosenbrock23{0, ADTypes.AutoFiniteDiff{Val{:forward}, Val{:forward}, Val{:hcentral}, Nothing, Nothing, Int64}, Nothing, typeof(OrdinaryDiffEqCore.DEFAULT_PRECS), Val{:forward}(), true, nothing, typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!)}, OrdinaryDiffEqRosenbrock.Rodas5P{0, ADTypes.AutoFiniteDiff{Val{:forward}, Val{:forward}, Val{:hcentral}, Nothing, Nothing, Int64}, Nothing, typeof(OrdinaryDiffEqCore.DEFAULT_PRECS), Val{:forward}(), true, nothing, typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!)}, OrdinaryDiffEqBDF.FBDF{5, 0, ADTypes.AutoFiniteDiff{Val{:forward}, Val{:forward}, Val{:hcentral}, Nothing, Nothing, Int64}, Nothing, OrdinaryDiffEqNonlinearSolve.NLNewton{Rational{Int64}, Rational{Int64}, Rational{Int64}, Nothing}, typeof(OrdinaryDiffEqCore.DEFAULT_PRECS), Val{:forward}(), true, nothing, Nothing, Nothing, typeof(OrdinaryDiffEqCore.trivial_limiter!)}, OrdinaryDiffEqBDF.FBDF{5, 0, ADTypes.AutoFiniteDiff{Val{:forward}, Val{:forward}, Val{:hcentral}, Nothing, Nothing, Int64}, LinearSolve.KrylovJL{typeof(Krylov.gmres!), Int64, Nothing, Tuple{}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}, OrdinaryDiffEqNonlinearSolve.NLNewton{Rational{Int64}, Rational{Int64}, Rational{Int64}, Nothing}, typeof(OrdinaryDiffEqCore.DEFAULT_PRECS), Val{:forward}(), true, nothing, Nothing, Nothing, typeof(OrdinaryDiffEqCore.trivial_limiter!)}}, Rational{Int64}, Int64}}, OrdinaryDiffEqCore.InterpolationData{SciMLBase.ODEFunction{false, SciMLBase.AutoSpecialize, Main.var"##231".var"#1#2", LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, Vector{Float64}, Vector{Float64}, Vector{Vector{Float64}}, Vector{Int64}, OrdinaryDiffEqCore.DefaultCache{OrdinaryDiffEqTsit5.Tsit5ConstantCache, OrdinaryDiffEqVerner.Vern7ConstantCache, OrdinaryDiffEqRosenbrock.Rosenbrock23ConstantCache{Float64, SciMLBase.TimeDerivativeWrapper{false, SciMLBase.ODEFunction{false, SciMLBase.AutoSpecialize, Main.var"##231".var"#1#2", LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, Float64, SciMLBase.NullParameters}, SciMLBase.UDerivativeWrapper{false, SciMLBase.ODEFunction{false, SciMLBase.AutoSpecialize, Main.var"##231".var"#1#2", LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, Float64, SciMLBase.NullParameters}, Float64, Float64, Nothing, ADTypes.AutoFiniteDiff{Val{:forward}, Val{:forward}, Val{:hcentral}, Nothing, Nothing, Int64}}, OrdinaryDiffEqRosenbrock.RosenbrockCombinedConstantCache{SciMLBase.TimeDerivativeWrapper{false, SciMLBase.ODEFunction{false, SciMLBase.AutoSpecialize, Main.var"##231".var"#1#2", LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, Float64, SciMLBase.NullParameters}, SciMLBase.UDerivativeWrapper{false, SciMLBase.ODEFunction{false, SciMLBase.AutoSpecialize, Main.var"##231".var"#1#2", LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, Float64, SciMLBase.NullParameters}, OrdinaryDiffEqRosenbrock.RodasTableau{Float64, Float64}, Float64, Float64, Nothing, ADTypes.AutoFiniteDiff{Val{:forward}, Val{:forward}, Val{:hcentral}, Nothing, Nothing, Int64}}, OrdinaryDiffEqBDF.FBDFConstantCache{5, OrdinaryDiffEqNonlinearSolve.NLSolver{OrdinaryDiffEqNonlinearSolve.NLNewton{Rational{Int64}, Rational{Int64}, Rational{Int64}, Nothing}, false, Float64, Rational{Int64}, Nothing, Float64, OrdinaryDiffEqNonlinearSolve.NLNewtonConstantCache{Float64, Float64, Float64, Float64, SciMLBase.UDerivativeWrapper{false, SciMLBase.ODEFunction{false, SciMLBase.AutoSpecialize, Main.var"##231".var"#1#2", LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, Float64, SciMLBase.NullParameters}}, Float64}, Vector{Float64}, Float64, Matrix{Float64}, Matrix{Float64}, StaticArraysCore.SMatrix{5, 6, Rational{Int64}, 30}, Float64, Vector{Float64}, Vector{Float64}}, OrdinaryDiffEqBDF.FBDFConstantCache{5, OrdinaryDiffEqNonlinearSolve.NLSolver{OrdinaryDiffEqNonlinearSolve.NLNewton{Rational{Int64}, Rational{Int64}, Rational{Int64}, Nothing}, false, Float64, Rational{Int64}, Nothing, Float64, OrdinaryDiffEqNonlinearSolve.NLNewtonConstantCache{Float64, Float64, Float64, OrdinaryDiffEqDifferentiation.WOperator{false, Float64, LinearAlgebra.UniformScaling{Bool}, Float64, Float64, Nothing, Float64, OrdinaryDiffEqDifferentiation.JVPCache{Float64, OrdinaryDiffEqDifferentiation.var"#9#10"{SciMLBase.ODEFunction{false, SciMLBase.AutoSpecialize, Main.var"##231".var"#1#2", LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, Float64, ADTypes.AutoFiniteDiff{Val{:forward}, Val{:forward}, Val{:hcentral}, Nothing, Nothing, Int64}, DifferentiationInterfaceFiniteDiffExt.FiniteDiffTwoArgPushforwardPrep{Nothing, Nothing, Float64, Float64, Int64}}, SciMLBase.ODEFunction{false, SciMLBase.AutoSpecialize, Main.var"##231".var"#1#2", LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, Float64, Float64, SciMLBase.NullParameters, Float64}}, SciMLBase.UDerivativeWrapper{false, SciMLBase.ODEFunction{false, SciMLBase.AutoSpecialize, Main.var"##231".var"#1#2", LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, Float64, SciMLBase.NullParameters}}, Float64}, Vector{Float64}, Float64, Matrix{Float64}, Matrix{Float64}, StaticArraysCore.SMatrix{5, 6, Rational{Int64}, 30}, Float64, Vector{Float64}, Vector{Float64}}, Tuple{Float64, Float64, DataType, DataType, DataType, Float64, Float64, SciMLBase.ODEFunction{false, SciMLBase.AutoSpecialize, Main.var"##231".var"#1#2", LinearAlgebra.UniformScaling{Bool}, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, Nothing, typeof(SciMLBase.DEFAULT_OBSERVED), Nothing, Nothing, Nothing, Nothing}, Float64, Float64, Float64, SciMLBase.NullParameters, Bool, Val{false}}, OrdinaryDiffEqCore.AutoSwitchCache{Tuple{OrdinaryDiffEqTsit5.Tsit5{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}, OrdinaryDiffEqVerner.Vern7{typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!), Static.False}}, Tuple{OrdinaryDiffEqRosenbrock.Rosenbrock23{0, ADTypes.AutoFiniteDiff{Val{:forward}, Val{:forward}, Val{:hcentral}, Nothing, Nothing, Int64}, Nothing, typeof(OrdinaryDiffEqCore.DEFAULT_PRECS), Val{:forward}(), true, nothing, typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!)}, OrdinaryDiffEqRosenbrock.Rodas5P{0, ADTypes.AutoFiniteDiff{Val{:forward}, Val{:forward}, Val{:hcentral}, Nothing, Nothing, Int64}, Nothing, typeof(OrdinaryDiffEqCore.DEFAULT_PRECS), Val{:forward}(), true, nothing, typeof(OrdinaryDiffEqCore.trivial_limiter!), typeof(OrdinaryDiffEqCore.trivial_limiter!)}, OrdinaryDiffEqBDF.FBDF{5, 0, ADTypes.AutoFiniteDiff{Val{:forward}, Val{:forward}, Val{:hcentral}, Nothing, Nothing, Int64}, Nothing, OrdinaryDiffEqNonlinearSolve.NLNewton{Rational{Int64}, Rational{Int64}, Rational{Int64}, Nothing}, typeof(OrdinaryDiffEqCore.DEFAULT_PRECS), Val{:forward}(), true, nothing, Nothing, Nothing, typeof(OrdinaryDiffEqCore.trivial_limiter!)}, OrdinaryDiffEqBDF.FBDF{5, 0, ADTypes.AutoFiniteDiff{Val{:forward}, Val{:forward}, Val{:hcentral}, Nothing, Nothing, Int64}, LinearSolve.KrylovJL{typeof(Krylov.gmres!), Int64, Nothing, Tuple{}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}, OrdinaryDiffEqNonlinearSolve.NLNewton{Rational{Int64}, Rational{Int64}, Rational{Int64}, Nothing}, typeof(OrdinaryDiffEqCore.DEFAULT_PRECS), Val{:forward}(), true, nothing, Nothing, Nothing, typeof(OrdinaryDiffEqCore.trivial_limiter!)}}, Rational{Int64}, Int64}, Float64}, Nothing}, SciMLBase.DEStats, Vector{Int64}, Nothing, Nothing, Nothing}

Each element in the results is an ODE solution

sim[1]

retcode: Success
Interpolation: 3rd order Hermite
t: 5-element Vector{Float64}:
 0.0
 0.09978762825015643
 0.34618860282423985
 0.6785772553549279
 1.0
u: 5-element Vector{Float64}:
 0.10736652923400047
 0.11875160898753823
 0.1523071148905638
 0.2130680263705455
 0.2947856373966985

You can plot all the results at once

plot(sim, linealpha=0.4)

Solving an SDE with Different Parameters

using StochasticDiffEq
using Plots

function lotka_volterra!(du, u, p, t)
    du[1] = p[1] * u[1] - p[2] * u[1] * u[2]
    du[2] = -3 * u[2] + u[1] * u[2]
end

function g!(du, u, p, t)
    du[1] = p[3] * u[1]
    du[2] = p[4] * u[2]
end

p = [1.5, 1.0, 0.1, 0.1]
prob = SDEProblem(lotka_volterra!, g!, [1.0, 1.0], (0.0, 10.0), p)

SDEProblem with uType Vector{Float64} and tType Float64. In-place: true
Non-trivial mass matrix: false
timespan: (0.0, 10.0)
u0: 2-element Vector{Float64}:
 1.0
 1.0

function prob_func(prob, i, repeat)
    x = 0.3 * rand(2)
    remake(prob, p=[p[1:2]; x])
end

ensemble_prob = EnsembleProblem(prob, prob_func=prob_func)
sim = solve(ensemble_prob, SRIW1(), trajectories=10)

plot(sim, linealpha=0.5, color=:blue, idxs=(0, 1))
plot!(sim, linealpha=0.5, color=:red, idxs=(0, 2))

Show the distribution of the ensemble solutions

summ = EnsembleSummary(sim, 0:0.1:10)
plot(summ, fillalpha=0.5)

Ensemble simulations of Modelingtoolkit (MTK) models

Radioactive decay example

using ModelingToolkit
using OrdinaryDiffEq
using Plots

@independent_variables t
@variables c(t) = 1.0
@parameters λ = 1.0
D = Differential(t)
@mtkbuild sys = ODESystem([D(c) ~ -λ * c], t)
prob = ODEProblem(sys, [], (0.0, 2.0), []);
solve(prob)

retcode: Success
Interpolation: 3rd order Hermite
t: 8-element Vector{Float64}:
 0.0
 0.10001999200479662
 0.34208427066999536
 0.6553980136343391
 1.0312652525315806
 1.4709405856363595
 1.9659576669700232
 2.0
u: 8-element Vector{Vector{Float64}}:
 [1.0]
 [0.9048193287657775]
 [0.7102883621328676]
 [0.5192354400036404]
 [0.35655576576996556]
 [0.2297097907863828]
 [0.14002247272452764]
 [0.1353360028400881]

Use the symbolic interface to change the parameter(s).

function changemtkparam(prob, i, repeat)
    # Make a new copy of the parameter vector
    # Ensure the changed will not affect the original ODE problem
    remake(prob, p=[λ => i * 0.5])
end

eprob = EnsembleProblem(prob, prob_func=changemtkparam)
sim = solve(eprob, trajectories=10)
plot(sim, linealpha=0.5)

---

This notebook was generated using Literate.jl.