A super simple estimation of a function using its derivative.

poly-test.jl

# Set up environment
import Pkg; Pkg.activate(".")

# Imports
using Distributions
using Plots
using Polynomials
using Optim
using ForwardDiff

# Test the projection method on the sin function
f(x) = sin(x)
df(x) = cos(x)

lower, upper = -5, 5
K = 20 # Polynomial basis
T = 100
xs = range(lower, upper, length=T)
ys = f.(xs)
dys = df.(xs)

function fit_model(a, xs)
    poly = Polynomial(a)
    f_hat = poly.(xs)
    df_hat = (derivative(poly)).(xs)
    ℓ = sum((dys - df_hat) .^ 2)

    # p = plot(xs, ys)
    # plot!(p, xs, f_hat)
    # display(p)

    return ℓ
end

init = zeros(K)
init[1] = 0.5
init[2] = 0.25

target(x) = fit_model(x, xs)
function dtarget(G, x)
    return ForwardDiff.gradient!(G, target, x)
end

# opt = optimize(target, dtarget, init, SimulatedAnnealing(), Optim.Options(iterations=1_000))
# opt = optimize(target, dtarget, init, NelderMead(), Optim.Options(iterations=10_000))
opt = optimize(target, init, LBFGS(), Optim.Options(iterations=1_000))
# opt = optimize(target, dtarget, init, LBFGS(), Optim.Options(iterations=100_000))