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sindy-identification
GitHub基于PySINDy从时间序列数据中发现非线性动力学系统的控制方程。适用于具有稀疏表示的确定性轨迹数据，通过构建稀疏模型dx/dt=f(x)识别底层ODE，支持自定义函数库和阈值调优。
Trigger Scenarios
需要从时间序列数据中推导微分方程
系统具有稀疏动态特性且数据相对干净
目标是发现确定性系统的底层ODE
Install
npx skills add synthetic-sciences/openscience --skill sindy-identification -g -y
SKILL.md
Frontmatter
{
"name": "sindy-identification",
"tags": [
"SINDy",
"System Identification",
"Dynamical Systems",
"Equation Discovery",
"Sparse Regression"
],
"author": "Synthetic Sciences",
"license": "MIT",
"version": "1.0.0",
"category": "physics",
"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.",
"dependencies": [
"pysindy>=2.1.0",
"scipy>=1.11.0",
"numpy>=1.24.0",
"matplotlib>=3.7.0"
]
}
SINDy — Sparse Identification of Nonlinear Dynamics
Overview
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.
When to Use
- You have trajectory data and want to find the governing ODE
- System is expected to have a sparse representation (few active terms)
- Input variables are known (you measured the right state variables)
- Data is relatively clean (or you can denoise it)
Do NOT Use When
- You want arbitrary symbolic expressions (use
symbolic-regressionwith PySR) - You only have steady-state data (SINDy needs time derivatives)
- System is stochastic (SINDy assumes deterministic dynamics)
- You have PDE data (use PDE-FIND variant, not standard SINDy)
Installation
pip install pysindy
Core Workflows
1. Basic SINDy (Discover ODE from Data)
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), [1, 1, 1], t_eval=t_train, rtol=1e-10)
x_train = sol.y.T # shape (N, 3)
# Fit SINDy model
model = ps.SINDy(
feature_names=["x", "y", "z"],
optimizer=ps.STLSQ(threshold=0.1), # sparsity threshold
feature_library=ps.PolynomialLibrary(degree=2),
)
model.fit(x_train, t=dt)
model.print()
# Expected output:
# x' = -10.000 x + 10.000 y
# y' = 28.000 x + -1.000 y + -1.000 x z
# z' = -2.667 z + 1.000 x y
2. Choosing the Sparsity Threshold
# 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}")
3. Custom Function Libraries
# 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()
4. Noisy Data (Smoothed Differentiation)
# 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()
5. Validate: Simulate and Compare
# 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}")
6. Coefficient Extraction
# 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)}")
Key Parameters
| 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 |
Tips
- Start with polynomial degree 2 — most physics ODEs are low-order polynomial
- Threshold tuning is critical — too low = overfitting, too high = missing terms
- Clean data first — SINDy is sensitive to noise in derivatives
- Check the library — if the true terms aren't in the library, SINDy can't find them
- Validate by simulation — a good SINDy model should reproduce the trajectory
Troubleshooting
| 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) |
Version History
- e9844a4 Current 2026-07-11 17:32
Dependencies
-
required
pysindy>=2.1.0 -
required
scipy>=1.11.0 -
required
numpy>=1.24.0 -
required
matplotlib>=3.7.0
