Newton and Hessian-Free Newton-Krylov with Second Order Adjoint Sensitivity Analysis

In many cases it may be more optimal or more stable to fit using second order Newton-based optimization techniques. Since SciMLSensitivity.jl provides second order sensitivity analysis for fast Hessians and Hessian-vector products (via forward-over-reverse), we can utilize these in our neural/universal differential equation training processes.

sciml_train is set up to automatically use second order sensitivity analysis methods if a second order optimizer is requested via Optim.jl. Thus Newton and NewtonTrustRegion optimizers will use a second order Hessian-based optimization, while KrylovTrustRegion will utilize a Krylov-based method with Hessian-vector products (never forming the Hessian) for large parameter optimizations.

using Flux, DiffEqFlux, Optimization, OptimizationFlux, DifferentialEquations,
    Plots, Random, OptimizationOptimJL

u0 = Float32[2.0; 0.0]
datasize = 30
tspan = (0.0f0, 1.5f0)
tsteps = range(tspan[1], tspan[2], length = datasize)

function trueODEfunc(du, u, p, t)
    true_A = [-0.1 2.0; -2.0 -0.1]
    du .= ((u .^ 3)'true_A)'
end

prob_trueode = ODEProblem(trueODEfunc, u0, tspan)
ode_data = Array(solve(prob_trueode, Tsit5(), saveat = tsteps))

dudt2 = Flux.Chain(x -> x .^ 3,
    Flux.Dense(2, 50, tanh),
    Flux.Dense(50, 2))
prob_neuralode = NeuralODE(dudt2, tspan, Tsit5(), saveat = tsteps)

function predict_neuralode(p)
    Array(prob_neuralode(u0, p)[1])
end

function loss_neuralode(p)
    pred = predict_neuralode(p)
    loss = sum(abs2, ode_data .- pred)
    return loss, pred
end

# Callback function to observe training
list_plots = []
iter = 0
callback = function (p, l, pred; doplot = false)
    global list_plots, iter

    if iter == 0
        list_plots = []
    end
    iter += 1

    display(l)

    # plot current prediction against data
    plt = scatter(tsteps, ode_data[1, :], label = "data")
    scatter!(plt, tsteps, pred[1, :], label = "prediction")
    push!(list_plots, plt)
    if doplot
        display(plot(plt))
    end

    return l < 0.01
end

adtype = Optimization.AutoZygote()
optf = Optimization.OptimizationFunction((x, p) -> loss_neuralode(x), adtype)

optprob1 = Optimization.OptimizationProblem(optf, prob_neuralode.p)
pstart = Optimization.solve(optprob1, ADAM(0.01), callback = callback, maxiters = 100).u

optprob2 = Optimization.OptimizationProblem(optf, pstart)
pmin = Optimization.solve(optprob2, NewtonTrustRegion(), callback = callback,
    maxiters = 200)
pmin = Optimization.solve(optprob2, Optim.KrylovTrustRegion(), callback = callback,
    maxiters = 200)

u: 252-element Vector{Float32}:
  0.18882741
 -0.2668401
  0.21164912
 -0.2377658
  0.036435824
  0.12858436
  0.1953777
 -0.056883283
  0.23847732
  0.039548066
  ⋮
 -0.25166848
  0.2222557
 -0.06292838
 -0.1506017
  0.26450688
 -0.22468951
 -0.2934427
  0.0
  0.0

Note that we do not demonstrate Newton() because we have not found a single case where it is competitive with the other two methods. KrylovTrustRegion() is generally the fastest due to its use of Hessian-vector products.