How to plot time history and phase diagram of data-driven equation

[Aminofa70]

Dear Developers,
Thanks for you great package. I used your package for modeling a non-autonomous Duffing equation.
I want to plot the time-history and phase digram of the ode obtained from data here (res and pm) and compare it with data (here data is generated using ode duffing). I would appreciate it if you could guid me?

using DataDrivenDiffEq
using ModelingToolkit
using DifferentialEquations
using LinearAlgebra
using DataDrivenSparse
using Plots
gr()
# --- Duffing system with forcing ---
function duffing(u, p, t)
    du = similar(u)
    δ = 0.3 ; α = 1.0 ; β = 1.0 ;γ = 0.1 ; ω = 1.2
    du[1] = u[2]
    du[2] = -δ*u[2] - α*u[1] - β*u[1]^3 + γ*cos(ω*t)
    return du
end
u0 = [1.0; 0.0]
tspan = (0.0, 200.0)
dt = 0.01
prob = ODEProblem(duffing, u0, tspan)
sol = solve(prob, Tsit5(), saveat=dt)
X = sol[:, :]
DX = similar(X)
for (i, xi) in enumerate(eachcol(X))
    DX[:, i] = duffing(xi, [], sol.t[i])
end
t = sol.t
ddprob = ContinuousDataDrivenProblem(X, t, DX=DX)
@parameters t
@variables u[1:2]
u = collect(u)
Ψ = Basis([u; u[1]^3; 0.1*cos.(1.2*t)], u, independent_variable=t)  
λ = 1e-3
opt = STLSQ(λ)
res = solve(ddprob, Ψ, opt)
fieldnames(typeof(res))
basis_result = get_basis(res)
fieldnames(typeof(basis_result))
pm = get_parameter_map(basis_result)
println("Discovered parameters:")
for (k,v) in pm
    println(k, " => ", v)
end

[ChrisRackauckas-Claude]

Here's how to plot time-history and phase diagrams from your discovered equations:

using DataDrivenDiffEq, OrdinaryDiffEq, Plots

# After your sparse identification
# res = solve(ddprob, basis, STLSQ(...), ...)

# Extract discovered basis and parameters
discovered_basis = get_basis(res)
p_discovered = get_parameter_values(discovered_basis)

# Create ODEProblem from discovered equations
u0 = [1.0, 0.0]  # your initial conditions [x₀, ẋ₀]
tspan = (0.0, 100.0)
discovered_prob = ODEProblem(discovered_basis, u0, tspan, p_discovered)

# Solve the discovered ODE
discovered_sol = solve(discovered_prob, Tsit5(), saveat=0.1)

# Time-history plot
p1 = plot(original_sol, idxs=1, label="Original x(t)", linestyle=:dash)
plot!(p1, discovered_sol, idxs=1, label="Discovered x(t)")
xlabel!("Time")
ylabel!("x")
title!("Time History Comparison")

# Phase diagram (x vs ẋ)
p2 = plot(original_sol, idxs=(1,2), label="Original", linestyle=:dash)
plot!(p2, discovered_sol, idxs=(1,2), label="Discovered")
xlabel!("x")
ylabel!("ẋ")
title!("Phase Portrait")

plot(p1, p2, layout=(1,2), size=(900,400))

DataDrivenDiffEq also has a built-in plot recipe that shows the comparison automatically:

plot(res)  # Shows discovered vs original data with residual errors

The key functions are:

• get_basis(res) - extracts the discovered equations as a Basis object
• get_parameter_values(basis) - gets parameter values as a numeric vector suitable for OrdinaryDiffEq
• ODEProblem(basis, u0, tspan, p) - creates an ODEProblem directly from the basis

For a forced Duffing oscillator specifically, make sure your time vector matches between the discovered and original solutions for accurate comparison.

[Aminofa70]

Awesome. Super thanks. Works well.