pysindy.DiscreteSINDy

class pysindy.DiscreteSINDy(optimizer: BaseOptimizer | None = None, feature_library: BaseFeatureLibrary | None = None)

Sparse Identification of Nonlinear Dynamical Systems (SINDy) for discrete time systems.

Parameters:

• optimizer (optimizer object, optional) – Optimization method used to fit the SINDy model. This must be a class extending pysindy.optimizers.BaseOptimizer. The default is STLSQ.

• feature_library (feature library object, optional) – Feature library object used to specify candidate right-hand side features. This must be a class extending pysindy.feature_library.base.BaseFeatureLibrary. The default option is PolynomialLibrary.

Attributes:

• model (sklearn.multioutput.MultiOutputRegressor object) – The fitted SINDy model.

• n_input_features_ (int) – The total number of input features.

• n_output_features_ (int) – The total number of output features. This number is a function of self.n_input_features and the feature library being used.

• n_control_features_ (int) – The total number of control input features.

Examples

>>> import numpy as np
>>> import pysindy as ps
>>> import matplotlib.pyplot as plt
>>> def f(x):
>>>     return 3.6 * x * (1 - x)
>>> n_steps = 20
>>> eps = 0.001
>>> x_map = np.zeros((n_steps))
>>> x_map[0] = 0.5
>>> for i in range(1, n_steps):
>>>     x_map[i] = f(x_map[i - 1]) + eps * np.random.randn()
>>> x_train_map = x_map[:16]
>>> x_test_map = x_map[16:]
>>> model = ps.DiscreteSINDy()
>>> model.fit(x_train_map, t=1)
>>> model.print()
(x0)[k+1] = 0.006 1 + 3.581 x0[k] + -3.586 x0[k]^2
>>> model.coefficients()
>>> model.predict(x_test_map)
AxesArray([[0.8268863 ],
        [0.51884209],
        [0.89919392],
        [0.33532148]])
>>> model.score(x_test_map, t=1)
0.9998547755296847
>>> model.simulate(x0=0.5, t=20)
array([[0.5       ],
   [0.90037072],
   [0.32376537],
   [0.78980964],
   [0.59788467],
   [0.86556721],
   [0.41950949],
   [0.877508  ],
   [0.38763939],
   [0.85561537],
   [0.44528986],
   [0.88988815],
   [0.35351698],
   [0.8241012 ],
   [0.52224203],
   [0.89849516],
   [0.32914646],
   [0.79648209],
   [0.58382694],
   [0.87479097]])

num = 1000
N = 1000
N_drop = 500
r0 = 3.5
rs = r0 + np.arange(num) / num * (4 - r0)
xss = []
for r in rs:
    xs = []
    x = 0.5
    for n in range(N + N_drop):
        if n >= N_drop:
            xs = xs + [x]
        x = r * x * (1 - x)
    xss = xss + [xs]
plt.figure(figsize=(4, 4), dpi=100)
for ind in range(num):
    plt.plot(
    np.ones(N) * rs[ind], xss[ind], ",", alpha=0.1, c="black", rasterized=True
    )
plt.xlabel("$r$")
plt.ylabel("$x_n$")
plt.show()

>>> rs_train = [3.6, 3.7, 3.8, 3.9]
>>> xs_train = [np.array(xss[np.where(np.array(rs) == r)[0][0]]) for r in rs_train]
>>> feature_lib = ps.PolynomialLibrary(degree=3, include_bias=True)
>>> parameter_lib = ps.PolynomialLibrary(degree=1, include_bias=True)
>>> lib = ps.ParameterizedLibrary(
>>>     feature_library=feature_lib,
>>>     parameter_library=parameter_lib,
>>>     num_features=1,
>>>     num_parameters=1,
>>> )
>>> opt = ps.STLSQ(threshold=1e-1, normalize_columns=False)
>>> model = ps.DiscreteSINDy(feature_library=lib, optimizer=opt)
>>> model.fit(xs_train, u=rs_train, t=1, feature_names=["x", "r"])
>>> model.print()
(x)[k+1] = 1.000 r[k] x[k] + -1.000 r[k] x[k]^2

Methods

equations	Get the right hand sides of the DiscreteSINDy model equations.
fit	Fit a DiscreteSINDy model.
print	Print the DiscreteSINDy model equations.
score	Returns a score for the next state prediction produced by the model.
set_fit_request	Configure whether metadata should be requested to be passed to the fit method.
set_predict_request	Configure whether metadata should be requested to be passed to the predict method.
set_score_request	Configure whether metadata should be requested to be passed to the score method.
simulate	Simulate the DiscreteSINDy model forward in time.

Attributes

feature_library
optimizer
feature_names

equations(precision: int = 3) → list[str]

Get the right hand sides of the DiscreteSINDy model equations.

Parameters:

precision (int, optional (default 3)) – Number of decimal points to include for each coefficient in the equation.

Returns:

equations – List of strings representing the DiscreteSINDy model equations for each input feature.

Return type:

list of strings

fit(x, t, x_next=None, u=None, feature_names: list[str] | None = None)

Fit a DiscreteSINDy model.

Parameters:

• x (array-like or list of array-like, shape (n_samples, n_input_features)) – Training data of the current state of the system. If training data contains multiple trajectories, x should be a list containing data for each trajectory. Individual trajectories may contain different numbers of samples.

• t (float, numpy array of shape (n_samples,), or list of numpy arrays) – If t is a float, it specifies the timestep between each sample. If array-like, it specifies the time at which each sample was collected. In this case the values in t must be strictly increasing. In the case of multi-trajectory training data, t may also be a list of arrays containing the collection times for each individual trajectory.

• x_next (array-like or list of array-like, shape (n_samples, n_input_features), optional (default None)) – Optional data of the system forwarded by one time step. If not provided, the next will be computed by taking the training data by one time step. If x_next is provided, it must match the shape of the training data and these values will be used as the next state.

• u (array-like or list of array-like, shape (n_samples, n_control_features), optional (default None)) – Control variables/inputs. Include this variable to use sparse identification for nonlinear dynamical systems for control (SINDYc). If training data contains multiple trajectories (i.e. if x is a list of array-like), then u should be a list containing control variable data for each trajectory. Individual trajectories may contain different numbers of samples.

• feature_names (list of string, length n_input_features, optional) – Names for the input features (e.g. ['x', 'y', 'z']). If None, will use ['x0', 'x1', ...].

Returns:

self

Return type:

a fitted DiscreteSINDy instance

print(precision=3, **kwargs)

Print the DiscreteSINDy model equations.

Parameters:

• precision (int, optional (default 3)) – Precision to be used when printing out model coefficients.

• **kwargs (Additional keyword arguments passed to the builtin print function)

score(x, t, u=None, x_next=None, metric=<function r2_score>, **metric_kws)

Returns a score for the next state prediction produced by the model.

Parameters:

• x (array-like or list of array-like, shape (n_samples, n_input_features)) – Samples from which to make predictions.

• t (float, numpy array of shape (n_samples,), or list of numpy arrays) – Time step between samples or array of collection times. Optional, used to compute the time derivatives of the samples if x_dot is not provided.

• u (array-like or list of array-like, shape(n_samples, n_control_features), optional (default None)) – Control variables. If multiple_trajectories==True then u must be a list of control variable data from each trajectory. If the model was fit with control variables then u is not optional.

• metric (callable, optional) – Metric function with which to score the prediction. Default is the R^2 coefficient of determination. See Scikit-learn for more options.

• metric_kws (dict, optional) – Optional keyword arguments to pass to the metric function.

Returns:

score – Metric function value for the model prediction of x_next.

Return type:

float

set_fit_request(*, feature_names: bool | None | str = '$UNCHANGED$', t: bool | None | str = '$UNCHANGED$', u: bool | None | str = '$UNCHANGED$', x: bool | None | str = '$UNCHANGED$', x_next: bool | None | str = '$UNCHANGED$') → DiscreteSINDy

Configure whether metadata should be requested to be passed to the fit method.

Note that this method is only relevant when this estimator is used as a sub-estimator within a meta-estimator and metadata routing is enabled with enable_metadata_routing=True (see sklearn.set_config). Please check the User Guide on how the routing mechanism works.

The options for each parameter are:

• True: metadata is requested, and passed to fit if provided. The request is ignored if metadata is not provided.

• False: metadata is not requested and the meta-estimator will not pass it to fit.

• None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.

• str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

Added in version 1.3.

Parameters:

• feature_names (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for feature_names parameter in fit.

• t (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for t parameter in fit.

• u (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for u parameter in fit.

• x (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for x parameter in fit.

• x_next (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for x_next parameter in fit.

Returns:

self – The updated object.

Return type:

object

set_predict_request(*, u: bool | None | str = '$UNCHANGED$', x: bool | None | str = '$UNCHANGED$') → DiscreteSINDy

Configure whether metadata should be requested to be passed to the predict method.

Note that this method is only relevant when this estimator is used as a sub-estimator within a meta-estimator and metadata routing is enabled with enable_metadata_routing=True (see sklearn.set_config). Please check the User Guide on how the routing mechanism works.

The options for each parameter are:

• True: metadata is requested, and passed to predict if provided. The request is ignored if metadata is not provided.

• False: metadata is not requested and the meta-estimator will not pass it to predict.

• None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.

• str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

Added in version 1.3.

Parameters:

• u (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for u parameter in predict.

• x (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for x parameter in predict.

Returns:

self – The updated object.

Return type:

object

set_score_request(*, metric: bool | None | str = '$UNCHANGED$', t: bool | None | str = '$UNCHANGED$', u: bool | None | str = '$UNCHANGED$', x: bool | None | str = '$UNCHANGED$', x_next: bool | None | str = '$UNCHANGED$') → DiscreteSINDy

Configure whether metadata should be requested to be passed to the score method.

Note that this method is only relevant when this estimator is used as a sub-estimator within a meta-estimator and metadata routing is enabled with enable_metadata_routing=True (see sklearn.set_config). Please check the User Guide on how the routing mechanism works.

The options for each parameter are:

• True: metadata is requested, and passed to score if provided. The request is ignored if metadata is not provided.

• False: metadata is not requested and the meta-estimator will not pass it to score.

• None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.

• str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

Added in version 1.3.

Parameters:

• metric (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for metric parameter in score.

• t (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for t parameter in score.

• u (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for u parameter in score.

• x (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for x parameter in score.

• x_next (str, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED) – Metadata routing for x_next parameter in score.

Returns:

self – The updated object.

Return type:

object

simulate(x0, t, u=None, stop_condition=None)

Simulate the DiscreteSINDy model forward in time.

Parameters:

• x0 (numpy array, size [n_features]) – Initial condition from which to simulate.

• t (int) – Number of time steps to predict.

• u (list/array, optional (default None)) – Control inputs. A list (with len(u) == t) or array (with u.shape[0] == 1) giving the control inputs at each step.

• stop_condition (function object, optional) – If model is in discrete time, optional function that gives a stopping condition for stepping the simulation forward.

Returns:

x – Simulation results

Return type:

numpy array, shape (n_samples, n_features)