Source code for gemseo.problems.dataset.burgers

# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
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# Lesser General Public License for more details.
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# You should have received a copy of the GNU Lesser General Public License
# along with this program; if not, write to the Free Software Foundation,
# 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.

<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

[docs]
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