Source code for gemseo.problems.dataset.burgers

# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
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# Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
# Contributors:
#    INITIAL AUTHORS - initial API and implementation and/or initial
#                           documentation
#        :author: Syver Doving Agdestein
#    OTHER AUTHORS   - MACROSCOPIC CHANGES
r"""Burgers dataset.

This :class:`.Dataset` contains solutions to the Burgers' equation with
periodic boundary conditions on the interval :math:`[0, 2\pi]` for different
time steps:

.. math::

   u_t + u u_x = \nu u_{xx},

An analytical expression can be obtained for the solution, using the Cole-Hopf
transform:

.. math::

   u(t, x) = - 2 \nu \frac{\phi'}{\phi},

where :math:`\phi` is solution to the heat equation
:math:`\phi_t = \nu \phi_{xx}`.

This :class:`.Dataset` is based on a full-factorial
design of experiments. Each sample corresponds to a given time step :math:`t`,
while each feature corresponds to a given spatial point :math:`x`.

`More information about Burgers' equation
<https://en.wikipedia.org/wiki/Burgers%27_equation>`_
"""

from __future__ import annotations

from numpy import exp
from numpy import hstack
from numpy import linspace
from numpy import newaxis
from numpy import pi
from numpy import square

from gemseo.datasets.io_dataset import IODataset




def create_burgers_dataset(
    n_samples: int = 30,
    n_x: int = 501,
    fluid_viscosity: float = 0.1,
    categorize: bool = True,
) -> IODataset:
    """Burgers dataset parametrization.

    Args:
        n_samples: The number of samples.
        n_x: The number of spatial points.
        fluid_viscosity: The fluid viscosity.
        categorize: Whether to distinguish
            between the different groups of variables.

    Returns:
        The Burgers dataset.
    """
    time = linspace(0, 2, n_samples)[:, newaxis]
    space = linspace(0, 2 * pi, n_x)[newaxis, :]
    visc = fluid_viscosity

    alpha = space - 4 * time
    alpha_2 = square(alpha)
    beta = 4 * visc * (time + 1)
    gamma = space - 4 * time - 2 * pi
    gamma_2 = square(gamma)
    phi = exp(-alpha_2 / beta) + exp(-gamma_2 / beta)
    phi_deriv = -2 * alpha / beta * exp(-alpha_2 / beta)
    phi_deriv -= 2 * gamma / beta * exp(-gamma_2 / beta)
    u_t = -2 * visc / phi * phi_deriv

    if categorize:
        groups = {"t": IODataset.INPUT_GROUP, "u_t": IODataset.OUTPUT_GROUP}
    else:
        groups = None

    data = hstack([time, u_t])

    dataset = IODataset.from_array(data, ["t", "u_t"], {"t": 1, "u_t": n_x}, groups)
    dataset.name = "Burgers"
    dataset.misc["x"] = [[node] for node in space[0]]
    dataset.misc["nu"] = visc
    return dataset