# 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 DiffEqSensitivity.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 setup 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

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}:
-0.019309593
-0.117555045
0.07602171
0.23178332
-0.01313986
0.22991917
0.034613825
-0.1822314
0.0019270801
0.13566555
⋮
0.011424722
-0.32676128
0.18099813
-0.27681148
-0.081288196
-0.12527779
-0.12972249
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.