菜单
Sparse Identification of Nonlinear Dynamics (SINDy) — discover governing equations from time-series data. Builds sparse dynamical system models dx/dt = f(x) from measurements using PySINDy. Use when you have trajectory data and want to find the underlying ODE.
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| name | sindy-identification |
| description | Sparse Identification of Nonlinear Dynamics (SINDy) — discover governing equations from time-series data. Builds sparse dynamical system models dx/dt = f(x) from measurements using PySINDy. Use when you have trajectory data and want to find the underlying ODE. |
| category | physics |
| version | 1.0.0 |
| author | Synthetic Sciences |
| license | MIT |
| tags | ["SINDy","System Identification","Dynamical Systems","Equation Discovery","Sparse Regression"] |
| dependencies | ["pysindy>=2.1.0","scipy>=1.11.0","numpy>=1.24.0","matplotlib>=3.7.0"] |
Discover governing ODEs from time-series data using SINDy (Brunton et al., PNAS 2016). Given measurements of state variables x(t), SINDy identifies the sparse set of terms in a library of candidate functions that best describe dx/dt.
symbolic-regression with PySR)pip install pysindy
import numpy as np
import pysindy as ps
from scipy.integrate import solve_ivp
import matplotlib.pyplot as plt
# Generate training data from Lorenz system (ground truth)
def lorenz(t, y, sigma=10, rho=28, beta=8/3):
return [sigma*(y[1]-y[0]), y[0]*(rho-y[2])-y[1], y[0]*y[1]-beta*y[2]]
dt = 0.001
t_train = np.arange(0, 10, dt)
sol = solve_ivp(lorenz, (0, 10), [, , ], t_eval=t_train, rtol=)
x_train = sol.y.T
model = ps.SINDy(
feature_names=[, , ],
optimizer=ps.STLSQ(threshold=),
feature_library=ps.PolynomialLibrary(degree=),
)
model.fit(x_train, t=dt)
model.()
# Sweep thresholds and check model complexity vs error
thresholds = [0.001, 0.01, 0.05, 0.1, 0.5, 1.0]
for thresh in thresholds:
model_t = ps.SINDy(
optimizer=ps.STLSQ(threshold=thresh),
feature_library=ps.PolynomialLibrary(degree=2),
)
model_t.fit(x_train, t=dt)
complexity = model_t.complexity # number of nonzero terms
# Simulate and compute error
x_sim = model_t.simulate(x_train[0], t_train)
rmse = np.sqrt(np.mean((x_sim - x_train)**2))
print(f"threshold={thresh:.3f}: {complexity} terms, RMSE={rmse:.4f}")
# Include trigonometric functions for oscillatory systems
library = ps.PolynomialLibrary(degree=2) + ps.FourierLibrary(n_frequencies=3)
# Or build a custom library
import pysindy as ps
custom_library = ps.CustomLibrary(
library_functions=[
lambda x: x, # linear
lambda x: x**2, # quadratic
lambda x: np.sin(x), # sinusoidal
lambda x: np.cos(x),
],
function_names=[
lambda x: x,
lambda x: f"{x}^2",
lambda x: f"sin({x})",
lambda x: f"cos({x})",
]
)
model = ps.SINDy(
feature_library=custom_library,
optimizer=ps.STLSQ(threshold=0.1),
)
model.fit(x_train, t=dt)
model.print()
# Add noise
x_noisy = x_train + 0.1 * np.random.randn(*x_train.shape)
# Use smoothed finite differences for derivatives
model_noisy = ps.SINDy(
differentiation_method=ps.SmoothedFiniteDifference(),
optimizer=ps.STLSQ(threshold=0.2), # higher threshold for noisy data
feature_library=ps.PolynomialLibrary(degree=2),
)
model_noisy.fit(x_noisy, t=dt)
model_noisy.print()
# Simulate discovered model from same IC
x_sim = model.simulate(x_train[0], t_train)
fig, axes = plt.subplots(3, 1, figsize=(10, 8), sharex=True)
labels = ['x', 'y', 'z']
for i, ax in enumerate(axes):
ax.plot(t_train, x_train[:, i], 'b-', linewidth=0.5, label='Ground truth')
ax.plot(t_train, x_sim[:, i], 'r--', linewidth=0.5, label='SINDy model')
ax.set_ylabel(labels[i], fontsize=13)
ax.legend(loc='upper right')
ax.grid(True, alpha=0.3)
axes[-1].set_xlabel('Time [s]')
axes[0].set_title('SINDy Model vs Ground Truth')
plt.tight_layout()
plt.savefig('sindy_validation.png', dpi=150, bbox_inches='tight')
# Quantitative error
rmse = np.sqrt(np.mean((x_sim - x_train)**2, axis=0))
print(f"RMSE per variable: x={rmse[0]:.4f}, y={rmse[1]:.4f}, z={rmse[2]:.4f}")
# Get coefficient matrix (rows = equations, cols = library functions)
coefficients = model.coefficients()
feature_names = model.get_feature_names()
print("Discovered equations:")
for i, eq_name in enumerate(['dx/dt', 'dy/dt', 'dz/dt']):
terms = []
for j, name in enumerate(feature_names):
if abs(coefficients[i, j]) > 1e-10:
terms.append(f"{coefficients[i,j]:+.3f}·{name}")
print(f" {eq_name} = {' '.join(terms)}")
| Parameter | Default | Description |
|---|---|---|
threshold (STLSQ) | 0.1 | Sparsity threshold — coefficients below this are zeroed |
degree (PolynomialLibrary) | 2 | Maximum polynomial degree in library |
alpha (STLSQ) | 0.05 | Ridge regularization strength |
max_iter (STLSQ) | 20 | Maximum iterations for thresholding |
| Symptom | Fix |
|---|---|
| Too many terms discovered | Increase threshold |
| Missing terms | Decrease threshold or check library contains needed functions |
| Poor simulation accuracy | Data too noisy — use SmoothedFiniteDifference |
| Model diverges on simulation | Discovered model is unstable — check coefficient signs |
x_train wrong shape | Must be (n_samples, n_features), NOT (n_features, n_samples) |