Multiple Shooting

Docs: https://docs.sciml.ai/DiffEqFlux/dev/examples/multiple_shooting/

In Multiple Shooting, the training data is split into overlapping intervals. The solver (OptimizationPolyalgorithms.PolyOpt()) is trained on individual intervals. The results are stiched together.

This simple method assumes no noise in the data. A more robust version can be found at JuliaSimModelOptimizer.jl, which is a proprietary software.

using Lux
using ComponentArrays
using DiffEqFlux
using DiffEqFlux: group_ranges
using Optimization
using OptimizationPolyalgorithms
using OrdinaryDiffEq
using Plots
using Random
rng = Random.Xoshiro(0)

Random.Xoshiro(0xdb2fa90498613fdf, 0x48d73dc42d195740, 0x8c49bc52dc8a77ea, 0x1911b814c02405e8, 0x22a21880af5dc689)

Define initial conditions and time steps

datasize = 51
u0 = [2.0, 0.0]
tspan = (0.0, 5.0)
tsteps = range(tspan[begin], tspan[end], length = datasize)

0.0:0.1:5.0

True values

true_A = [-0.1 2.0; -2.0 -0.1]

2×2 Matrix{Float64}:
 -0.1   2.0
 -2.0  -0.1

Generate data from the true function: \(x^3 * A\)

function trueODEfunc!(du, u, p, t)
    du .= ((u.^3)'true_A)'
end
prob_trueode = ODEProblem(trueODEfunc!, u0, tspan)
ode_data = Array(solve(prob_trueode, Tsit5(), saveat = tsteps))

2×51 Matrix{Float64}:
 2.0  1.76453  0.666824  -0.580558  …  0.0306948  -0.138592  -0.302836
 0.0  1.4286   1.86579    1.80634      0.950209    0.941635   0.930954

Define the Neural Network using Lux.jl Notice the network is smaller than the first example.

nn = Lux.Chain(
    x -> x.^3,
    Lux.Dense(2, 16, tanh),
    Lux.Dense(16, 2)
)
p_init, st = Lux.setup(rng, nn) |> f64
ps = ComponentArray(p_init)
pd, pax = getdata(ps), getaxes(ps)

([-1.8019577264785767, -0.18273845314979553, 1.6776520013809204, 0.19449931383132935, 0.7557111978530884, 1.1159610748291016, -1.581186056137085, 1.7986798286437988, -0.36156967282295227, -1.9202053546905518  …  0.3558240234851837, -0.2906492352485657, 0.32653868198394775, 0.3687601387500763, -0.3538714349269867, 0.12959939241409302, 0.256054550409317, -0.20957911014556885, 0.10817152261734009, -0.20544955134391785], (Axis(layer_1 = ViewAxis(1:0, Shaped1DAxis((0,))), layer_2 = ViewAxis(1:48, Axis(weight = ViewAxis(1:32, ShapedAxis((16, 2))), bias = ViewAxis(33:48, Shaped1DAxis((16,))))), layer_3 = ViewAxis(49:82, Axis(weight = ViewAxis(1:32, ShapedAxis((2, 16))), bias = ViewAxis(33:34, Shaped1DAxis((2,)))))),))

Define the NeuralODE problem

neuralode = NeuralODE(nn, tspan, Tsit5(), saveat = tsteps)
prob_node = ODEProblem((u,p,t)->nn(u,p,st)[1], u0, tspan, ComponentArray(p_init))

ODEProblem with uType Vector{Float64} and tType Float64. In-place: false
Non-trivial mass matrix: false
timespan: (0.0, 5.0)
u0: 2-element Vector{Float64}:
 2.0
 0.0

Animate training process in the callback function

function plot_multiple_shoot(plt, preds, group_size)
	ranges = group_ranges(datasize, group_size)
	for (i, rg) in enumerate(ranges)
		plot!(plt, tsteps[rg], preds[i][1,:], markershape=:circle, label="Group $(i)")
	end
end

anim = Animation()
lossrecord=Float64[]
callback = function (state, l; doplot = true)
    if doplot
        plt = scatter(tsteps, ode_data[1,:], label = "Data")
        plot_multiple_shoot(plt, preds, group_size)
        frame(anim)
        push!(lossrecord, l)
    end
    return false
end

#9 (generic function with 1 method)

Parameters for Multiple Shooting

group_size = 3
continuity_term = 200  ## Penalty for discontinuity

function loss_function(data, pred)
    return sum(abs2, data .- pred)
end

l1, preds = multiple_shoot(ps, ode_data, tsteps, prob_node, loss_function, Tsit5(), group_size; continuity_term)

function loss_multiple_shooting(p)
    ps = ComponentArray(p, pax)

    loss, currpred = multiple_shoot(ps, ode_data, tsteps, prob_node, loss_function,
        Tsit5(), group_size; continuity_term)
    global preds = currpred
    return loss
end

loss_multiple_shooting (generic function with 1 method)

Solve the problem using OptimizationPolyalgorithms.PolyOpt().

adtype = Optimization.AutoZygote()
optf = Optimization.OptimizationFunction((x,p) -> loss_multiple_shooting(x), adtype)
optprob = Optimization.OptimizationProblem(optf, pd)
res_ms = Optimization.solve(optprob, PolyOpt(), callback = callback)

println("Loss is ", loss_multiple_shooting(res_ms.u)[1])

Loss is 1.6362929930673293

Visualize the fitting processes

mp4(anim, fps=15)

[ Info: Saved animation to /tmp/jl_9QBcN8RgFO.mp4

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This notebook was generated using Literate.jl.