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 DiffEqFlux, OrdinaryDiffEq, Flux, Optim, Plots

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 = FastChain((x, p) -> x.^3,
                  FastDense(2, 50, tanh),
                  FastDense(50, 2))
prob_neuralode = NeuralODE(dudt2, tspan, Tsit5(), saveat = tsteps)
p = prob_neuralode.p

function predict_neuralode(p)
  Array(prob_neuralode(u0, p))
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
cb = 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

pstart = DiffEqFlux.sciml_train(loss_neuralode, p, ADAM(0.01), cb=cb, maxiters = 100).minimizer
pmin = DiffEqFlux.sciml_train(loss_neuralode, pstart, NewtonTrustRegion(), cb=cb, maxiters = 200)
pmin = DiffEqFlux.sciml_train(loss_neuralode, pstart, Optim.KrylovTrustRegion(), cb=cb, maxiters = 200)

Note that we do note 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.