Difference equation with neural network

Someone can give me a research direction to find the time series w[i] so that the loss function sum(F.^2) is minimized for a known time series q.

# define a function to calculate the values of F for a given time series q, time series of 𝜭, and value of ζ
# 𝜭[i] is defined as 𝜭[i] = cumsum (w[i] * Δt[i])   where Δt[i] is the time period.

ζ = 1/(4*pi)
function calculate_F(q, ζ, 𝜭)
    F = Float64[]
    for i in 1 : length(q) - 2
        value = q[i+2] - 2*exp(-ζ*𝜭[i]) * cos(𝜭[i] ) * q[i+1] + exp(-2*ζ*𝜭[i]) * q[i]
        push!(F, value)
    return F