Parameters in DiffEqFlux

I’m having trouble understanding how to enter the trainable parameters into DiffEqFlux.

In order to understand how DiffEqFlux works, I set myself the following exercise: Replace p = [1.5, 1.0, 3.0, 1.0] in the example of optimizing an ODE, with a neural network that learns the parameters as a function of an input vector (say, u0, but it could be anything).

(This is, of course, an impractical complication, the goal is that it prove an instructive exercise.)

The problem I’m running into is that I can’t figure out how to enter the resulting parameters into DiffEqFlux.sciml_train.

My code is as follows:

#Copied from the manual
using DifferentialEquations, Flux, Optim, DiffEqFlux, DiffEqSensitivity, Plots

function lotka_volterra!(du, u, p, t)
  x, y = u
  α, β, δ, γ = p
  du[1] = dx = α*x - β*x*y
  du[2] = dy = -δ*y + γ*x*y

#My neural net (by hand for the sake of the exercise):
W1 = rand(5,2)
b1 = rand(5)
W2 = rand(4,5)
b2 = rand(4)
model(x) = relu.(W2*relu.(W1*x + b1) + b2)

#Copied from the manual
tspan = (0.0, 10.0)
tsteps = 0.0:0.1:10.0

#Copied from the manual, but with p replaced by model(u0)).
prob = ODEProblem(lotka_volterra!, u0, tspan, model(u0))
sol = solve(prob, Tsit5())

using Plots
plot(sol) #Generates a plot using the model.

function loss()
  sol = solve(prob, Tsit5(), p=#=Can't see what goes here=#, saveat = tsteps)
  loss = sum(abs2, sol.-1)
  return loss, sol

callback = function (p, l, pred)
  plt = plot(pred, ylim = (0, 6))
  return false

result_ode = DiffEqFlux.sciml_train(loss,
                                     #=params(W1,W2,b1,b2) returns error=#,
                                    cb = callback,
                                    maxiters = 100)

Anyway, I’m kinda at a loss here for how to handle the parameters. I can’t put params(W1,W2,b1,2) into the sciml_train function, since it returns an error:

ERROR: MethodError: no method matching copy(::Zygote.Params)
Closest candidates are:
  copy(::Expr) at expr.jl:36
  copy(::Core.CodeInfo) at expr.jl:64
  copy(::BitSet) at bitset.jl:46
 [1] sciml_train(::Function, ::Zygote.Params, ::ADAM, ::Base.Iterators.Cycle{Tuple{DiffEqFlux.NullData}}; cb::Function, maxiters::Int64, progress::Bool, save_best::Bool) at C:\Users\MattProgramming\.juliapro\JuliaPro_v1.4.2-1\packages\DiffEqFlux\FZMwP\src\train.jl:77
 [2] top-level scope at none:0

And if I put in the parameters explicitly (e.g., using W1 for the parameter), nothing trains.

(I think I may be able to solve this problem as a neural differential equation, by chaining lotka_volterra into the neural differential equation, but this problem doesn’t seem, in principle, any more complicated than the manual example. Except I can’t figure out how to tell it which parameters to use.)

The parameters need to be an array. If you want to use an array that has a bit more structure, I’d suggest using ComponentArrays.jl, which has an example on this:

Or if you want to have multiple different arrays for the parameters, you can use the Flux style for the optimization demonstrated at:

(However, note that the Flux style does limit the available optimizers)

Thank you! This is very helpful.