I am getting a problem with trying to include time dependant libraries in the when using sindy

[munozdjp]

describing the general problem

I can not get an approximation of my system when using time dependant parameters:

I want to predict my system

I have tried the following:

import os
import sys
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint, solve_ivp
import pysindy as ps
from pysindy.feature_library import ConcatLibrary

# Define the system of equations
def system(t, x):
    dxdt = t**2 * x[0] - x[1] - x[0]**3 * t**3 + x[0] * x[1]**2
    dydt = x[0] * t + x[1] * t**2 + t**4 * x[1] - x[1] * (x[0]**2 + x[1]**2)
    return [dxdt, dydt]

# Initial conditions and time span
x0 = 1
y0 = 1
xoinitial = [x0, y0]
tspan = (0, 10)
t_eval = np.arange(0, 10, 0.01)  # Time points for evaluation

# Integrate the system
sol = solve_ivp(system, tspan, xoinitial, t_eval=t_eval)

# Prepare data for SINDy
x_train = sol.y.T
 # Stack x and y
t_train = sol.t

def compute_derivatives_analytical(x_solutions, t_eval):
    xdot = np.zeros_like(x_solutions.T)

    # Loop through each time point and evaluate the derivative
    for i, t in enumerate(t_eval):
        xdot[i] = system(
            t,
            x_solutions[:, i]
        )

    return xdot

# Compute the derivatives analytically
xdot = compute_derivatives_analytical(sol.y, t_eval)

# Create feature libraries (polynomial + time)
polynomial_library = ps.PolynomialLibrary(degree=3)
custom_library = ps.CustomLibrary(library_functions=[lambda x, t: t], function_names=["t"])

# Combine polynomial and time-dependent libraries
combined_library = ConcatLibrary([polynomial_library, custom_library])

# Initialize SINDy model with time-dependent features
differentiation_method = ps.FiniteDifference(order=2)
optimizer = ps.STLSQ(threshold=0.1)

sindy_model = ps.SINDy(
    feature_library=combined_library,
    optimizer=optimizer,
    feature_names=["x0", "x1"]
)

# Fit SINDy model
sindy_model.fit(x=x_train, x_dot=xdot, t=t_eval)

# Print the discovered SINDy model
sindy_model.print()

# Predict using SINDy model (optional)
x_pred = sindy_model.simulate(xoinitial, t_eval)

# Plot comparison of original vs predicted
plt.figure(figsize=(10, 6))
plt.plot(t_eval, sol.y[0], label='Original x0', linestyle='--')
plt.plot(t_eval, x_pred[:, 0], label='Predicted x0')
plt.plot(t_eval, sol.y[1], label='Original x1', linestyle='--')
plt.plot(t_eval, x_pred[:, 1], label='Predicted x1')
plt.title('SINDy Model vs Original Hopf Model')
plt.xlabel('Time (t)')
plt.ylabel('x0, x1 values')
plt.grid(True)
plt.legend()
plt.show()

Additional context

however I get a weird error when trying to print the system and it seems i do not have equation when using sindy.