mdo_function module¶

Base class to describe a function.

class gemseo.core.mdofunctions.mdo_function.MDOFunction(func, name, f_type=FunctionType.NONE, jac=None, expr='', input_names=None, dim=0, output_names=None, force_real=False, special_repr='', original_name='', expects_normalized_inputs=False)[source]¶

Bases: Serializable

The standard definition of an array-based function with algebraic operations.

MDOFunction is the key class to define the objective function, the constraints and the observables of an OptimizationProblem.

an MDOFunction is initialized from an optional callable and a name, e.g. func = MDOFunction(lambda x: 2*x, "my_function").

Note

The callable can be set to None when the user does not want to use a callable but a database to browse for the output vector corresponding to an input vector (see MDOFunction.set_pt_from_database()).

The following information can also be provided at initialization:

the type of the function, e.g. f_type="obj" if the function will be used as an objective (see MDOFunction.FunctionType),
the function computing the Jacobian matrix, e.g. jac=lambda x: array([2.]),
the literal expression to be used for the string representation of the object, e.g. expr="2*x",
the names of the inputs and outputs of the function, e.g. input_names=["x"] and output_names=["y"].

Warning

For the literal expression, do not use “f(x) = 2*x” nor “f = 2*x” but “2*x”. The other elements will be added automatically in the string representation of the function based on the name of the function and the names of its inputs.

After the initialization, all of these arguments can be overloaded with setters, e.g. MDOFunction.input_names.

The original function and Jacobian function can be accessed with the properties MDOFunction.func and MDOFunction.jac.

an MDOFunction is callable: output = func(array([3.])) # expected: array([6.]).

Elementary operations can be performed with MDOFunction instances: addition (func = func1 + func2 or func = func1 + offset), subtraction (func = func1 - func2 or func = func1 - offset), multiplication (func = func1 * func2 or func = func1 * factor) and opposite (func = -func1). It is also possible to build an MDOFunction as a concatenation of MDOFunction objects: func = MDOFunction.concatenate([func1, func2, func3], "my_func_123").

Moreover, an MDOFunction can be approximated with either a first-order or second-order Taylor polynomial at a given input vector, using respectively MDOFunction.linear_approximation() and quadratic_approx(); such an approximation is also an MDOFunction.

Lastly, the user can check the Jacobian function by means of approximation methods (see MDOFunction.check_grad()).

Initialize self. See help(type(self)) for accurate signature.

Parameters:

func (WrappedFunctionType | None) – The original function to be actually called. If None, the function will not have an original function.
name (str) – The name of the function.
f_type (FunctionType) –
The type of the function.

By default it is set to “”.
jac (WrappedJacobianType | None) – The original Jacobian function to be actually called. If None, the function will not have an original Jacobian function.
expr (str) –
The expression of the function, e.g. “2*x”, if any.

By default it is set to “”.
input_names (Iterable[str] | None) – The names of the inputs of the function. If None, the inputs of the function will have no names.
dim (int) –
The dimension of the output space of the function. If 0, the dimension of the output space of the function will be deduced from the evaluation of the function.

By default it is set to 0.
output_names (Iterable[str] | None) – The names of the outputs of the function. If None, the outputs of the function will have no names.
force_real (bool) –
Whether to cast the output values to real.

By default it is set to False.
special_repr (str) –
The string representation of the function. If empty, use default_repr().

By default it is set to “”.
original_name (str) –
The original name of the function. If empty, use the same name than the name input.

By default it is set to “”.
expects_normalized_inputs (bool) –
Whether the function expects normalized inputs.

By default it is set to False.

class ApproximationMode(value)¶

Bases: StrEnum

The approximation derivation modes.

CENTERED_DIFFERENCES = 'centered_differences'¶: The centered differences method used to approximate the Jacobians by perturbing each variable with a small real number.

COMPLEX_STEP = 'complex_step'¶: The complex step method used to approximate the Jacobians by perturbing each variable with a small complex number.

FINITE_DIFFERENCES = 'finite_differences'¶: The finite differences method used to approximate the Jacobians by perturbing each variable with a small real number.

class ConstraintType(value)[source]¶

Bases: StrEnum

The type of constraint.

EQ = 'eq'¶: The type of function for equality constraint.

INEQ = 'ineq'¶: The type of function for inequality constraint.

class FunctionType(value)¶

Bases: StrEnum

An enumeration.

EQ = 'eq'¶

INEQ = 'ineq'¶

NONE = ''¶

OBJ = 'obj'¶

OBS = 'obs'¶

check_grad(x_vect, approximation_mode=ApproximationMode.FINITE_DIFFERENCES, step=1e-06, error_max=1e-08)[source]¶

Check the gradients of the function.

Parameters:

x_vect (ndarray[Any, dtype[number]]) – The vector at which the function is checked.
approximation_mode (ApproximationMode) –
The approximation mode.

By default it is set to “finite_differences”.
step (float) –
The step for the approximation of the gradients.

By default it is set to 1e-06.
error_max (float) –
The maximum value of the error.

By default it is set to 1e-08.

Raises:

ValueError – Either if the approximation method is unknown, if the shapes of the analytical and approximated Jacobian matrices are inconsistent or if the analytical gradients are wrong.

Return type:

None

evaluate(x_vect)[source]¶

Evaluate the function and store the dimension of the output space.

Parameters:: x_vect (ndarray[Any, dtype[number]]) – The value of the inputs of the function.
Returns:: The value of the output of the function.
Return type:: ndarray[Any, dtype[number]] | Number

static filt_0(arr, floor_value=1e-06)[source]¶

Set the non-significant components of a vector to zero.

The component of a vector is non-significant if its absolute value is lower than a threshold.

Parameters:

arr (ndarray[Any, dtype[number]]) – The original vector.
floor_value (float) –
The threshold.

By default it is set to 1e-06.

Returns:

The original vector whose non-significant components have been set at zero.

Return type:

ndarray[Any, dtype[number]]

static from_pickle(file_path)[source]¶

Deserialize a function from a file.

Parameters:: file_path (str | Path) – The path to the file containing the function.
Returns:: The function instance.
Return type:: MDOFunction

classmethod generate_input_names(input_dim, input_names=None)[source]¶

Generate the names of the inputs of the function.

Parameters:

input_dim (int) – The dimension of the input space of the function.
input_names (Sequence[str] | None) – The initial names of the inputs of the function. If there is only one name, e.g. ["var"], use this name as a base name and generate the names of the inputs, e.g. ["var!0", "var!1", "var!2"] if the dimension of the input space is equal to 3. If None, use "x" as a base name and generate the names of the inputs, i.e. ["x!0", "x!1", "x!2"].

Returns:

The names of the inputs of the function.

Return type:

Sequence[str]

get_indexed_name(index)[source]¶

Return the name of function component.

Parameters:: index (int) – The index of the function component.
Returns:: The name of the function component.
Return type:: str

static init_from_dict_repr(**attributes)[source]¶

Initialize a new function.

This is typically used for deserialization.

Parameters:: **attributes (Any) – The values of the serializable attributes listed in MDOFunction.DICT_REPR_ATTR.
Returns:: A function initialized from the provided data.
Raises:: ValueError – If the name of an argument is not in MDOFunction.DICT_REPR_ATTR.
Return type:: MDOFunction

is_constraint()[source]¶

Check if the function is a constraint.

The type of a constraint function is either ‘eq’ or ‘ineq’.

Returns:: Whether the function is a constraint.
Return type:: bool

offset(value)[source]¶

Add an offset value to the function.

Parameters:: value (ndarray[Any, dtype[number]] | Number) – The offset value.
Returns:: The offset function.
Return type:: MDOFunction

static rel_err(a_vect, b_vect, error_max)[source]¶

Compute the 2-norm of the difference between two vectors.

Normalize it with the 2-norm of the reference vector if the latter is greater than the maximal error.

Parameters:

a_vect (ndarray[Any, dtype[number]]) – A first vector.
b_vect (ndarray[Any, dtype[number]]) – A second vector, used as a reference.
error_max (float) – The maximum value of the error.

Returns:

The difference between two vectors, normalized if required.

Return type:

float

set_pt_from_database(database, design_space, normalize=False, jac=True, x_tolerance=1e-10)[source]¶

Set the original function and Jacobian function from a database.

For a given input vector, the method MDOFunction.func() will return either the output vector stored in the database if the input vector is present or None. The same for the method MDOFunction.jac().

Parameters:

database (Database) – The database to read.
design_space (DesignSpace) – The design space used for normalization.
normalize (bool) –
If True, the values of the inputs are unnormalized before call.

By default it is set to False.
jac (bool) –
If True, a Jacobian pointer is also generated.

By default it is set to True.
x_tolerance (float) –
The tolerance on the distance between inputs.

By default it is set to 1e-10.

Return type:

None

to_dict()[source]¶

Create a dictionary representation of the function.

This is used for serialization. The pointers to the functions are removed.

Returns:: Some attributes of the function indexed by their names. See MDOFunction.DICT_REPR_ATTR.
Return type:: dict[str, str | int | list[str]]

to_pickle(file_path)[source]¶

Serialize the function and store it in a file.

Parameters:: file_path (str | Path) – The path to the file to store the function.
Return type:: None

COEFF_FORMAT_1D: str = '{:.2e}'¶: The format to be applied to a number when represented in a vector.

COEFF_FORMAT_ND: str = '{: .2e}'¶: The format to be applied to a number when represented in a matrix.

DEFAULT_BASE_INPUT_NAME: str = 'x'¶: The default base name for the inputs.

DICT_REPR_ATTR: ClassVar[list[str]] = ['name', 'f_type', 'expr', 'input_names', 'dim', 'special_repr', 'output_names']¶: The names of the attributes to be serialized.

INDEX_PREFIX: str = '!'¶: The character used to separate a name base and a prefix, e.g. "x!1.

activate_counters: ClassVar[bool] = True¶: Whether to count the number of function evaluations.

property default_repr: str¶: The default string representation of the function.

property dim: int¶: The dimension of the output space of the function.

property expects_normalized_inputs: bool¶: Whether the function expects normalized inputs.

property expr: str¶: The expression of the function, e.g. “2*x”.

force_real: bool¶: Whether to cast the results to real value.

property func: Callable[[ndarray[Any, dtype[number]]], ndarray[Any, dtype[number]] | Number]¶: The function to be evaluated from a given input vector.

has_default_name: bool¶: Whether the name has been set with a default value.

property has_jac: bool¶

Check if the function has an implemented Jacobian function.

Returns:: Whether the function has an implemented Jacobian function.

property input_names: list[str]¶

The names of the inputs of the function.

Use a copy of the original names.

property jac: Callable[[ndarray[Any, dtype[number]]], ndarray[Any, dtype[number]]]¶: The Jacobian function to be evaluated from a given input vector.

last_eval: OutputType | None¶

The value of the function output at the last evaluation.

None if it has not yet been evaluated.

property n_calls: int¶

The number of times the function has been evaluated.

This count is both multiprocess- and multithread-safe, thanks to the locking process used by MDOFunction.evaluate().

property name: str¶: The name of the function.

property original_name: str¶: The original name of the function.

property output_names: list[str]¶

The names of the outputs of the function.

Use a copy of the original names.

special_repr: str¶: The string representation of the function overloading its default string ones.

Examples using MDOFunction¶

Analytical test case # 1

Analytical test case # 2

Analytical test case # 3

Change the seed of a DOE

Scaling

Post-process an optimization problem

Save an optimization problem for post-processing