Compute the Jacobian of a discipline analytically

In this example, we will compute the Jacobians of some outputs of an Discipline with respect to some inputs, based on its analytical derivatives.

from __future__ import annotations

from numpy import array

from gemseo.disciplines.analytic import AnalyticDiscipline

First, we create a discipline, e.g. an AnalyticDiscipline:

discipline = AnalyticDiscipline({"y": "a**2+b", "z": "a**3+b**2"})

We can execute it with its default input values:

discipline.execute()
discipline.local_data

{'a': array([0.]), 'b': array([0.]), 'y': array([0.]), 'z': array([0.])}

or with custom ones:

discipline.execute({"a": array([1.0])})
discipline.local_data

{'a': array([1.]), 'b': array([0.]), 'y': array([1.]), 'z': array([1.])}

Then, we use the method Discipline.linearize() to compute the derivatives:

jacobian_data = discipline.linearize()
jacobian_data

{}

There is no Jacobian data because we need to set the input variables with respect to which to compute the Jacobian of the output ones. For that, we use the method add_differentiated_inputs(). We also need to set these output variables: with the method add_differentiated_outputs(). For instance, we may want to only compute the derivative of "z" with respect to "a":

discipline.add_differentiated_inputs(["a"])
discipline.add_differentiated_outputs(["z"])
jacobian_data = discipline.linearize()
jacobian_data

defaultdict(<function _default_dict_factory at 0x72a513d149a0>, {'z': defaultdict(None, {'a': array([[0.]])})})

By default, GEMSEO uses Discipline.default_input_data as input data for which to compute the Jacobian on. We can change them with input_data:

jacobian_data = discipline.linearize({"a": array([1.0])})
jacobian_data

defaultdict(<function _default_dict_factory at 0x72a513d149a0>, {'z': defaultdict(None, {'a': array([[3.]])})})

We can also force the discipline to compute the derivatives of all the outputs with respect to all the inputs:

jacobian_data = discipline.linearize(compute_all_jacobians=True)
jacobian_data

defaultdict(<function _default_dict_factory at 0x72a513d149a0>, {'y': defaultdict(None, {'a': array([[0.]]), 'b': array([[1.]])}), 'z': defaultdict(None, {'a': array([[0.]]), 'b': array([[0.]])})})