76 lines
2.3 KiB
Python
76 lines
2.3 KiB
Python
"""Formulation of the advection problem."""
|
|
|
|
import torch
|
|
from ... import Condition
|
|
from ...problem import SpatialProblem, TimeDependentProblem
|
|
from ...equation import Equation, Advection
|
|
from ...utils import check_consistency
|
|
from ...domain import CartesianDomain
|
|
|
|
|
|
def initial_condition(input_, output_):
|
|
"""
|
|
Implementation of the initial condition.
|
|
|
|
:param LabelTensor input_: Input data of the problem.
|
|
:param LabelTensor output_: Output data of the problem.
|
|
:return: The residual of the initial condition.
|
|
:rtype: LabelTensor
|
|
"""
|
|
return output_ - torch.sin(input_.extract("x"))
|
|
|
|
|
|
class AdvectionProblem(SpatialProblem, TimeDependentProblem):
|
|
r"""
|
|
Implementation of the advection problem in the spatial interval
|
|
:math:`[0, 2 \pi]` and temporal interval :math:`[0, 1]`.
|
|
|
|
.. seealso::
|
|
|
|
**Original reference**: Wang, Sifan, et al. *An expert's guide to
|
|
training physics-informed neural networks*.
|
|
arXiv preprint arXiv:2308.08468 (2023).
|
|
DOI: `arXiv:2308.08468 <https://arxiv.org/abs/2308.08468>`_.
|
|
|
|
:Example:
|
|
>>> problem = AdvectionProblem(c=1.0)
|
|
"""
|
|
|
|
output_variables = ["u"]
|
|
spatial_domain = CartesianDomain({"x": [0, 2 * torch.pi]})
|
|
temporal_domain = CartesianDomain({"t": [0, 1]})
|
|
|
|
domains = {
|
|
"D": CartesianDomain({"x": [0, 2 * torch.pi], "t": [0, 1]}),
|
|
"t0": CartesianDomain({"x": [0, 2 * torch.pi], "t": 0.0}),
|
|
}
|
|
|
|
conditions = {
|
|
"t0": Condition(domain="t0", equation=Equation(initial_condition)),
|
|
}
|
|
|
|
def __init__(self, c=1.0):
|
|
"""
|
|
Initialization of the :class:`AdvectionProblem`.
|
|
|
|
:param c: The advection velocity parameter.
|
|
:type c: float | int
|
|
"""
|
|
super().__init__()
|
|
check_consistency(c, (float, int))
|
|
self.c = c
|
|
|
|
self.conditions["D"] = Condition(domain="D", equation=Advection(self.c))
|
|
|
|
def solution(self, pts):
|
|
"""
|
|
Implementation of the analytical solution of the advection problem.
|
|
|
|
:param LabelTensor pts: Points where the solution is evaluated.
|
|
:return: The analytical solution of the advection problem.
|
|
:rtype: LabelTensor
|
|
"""
|
|
sol = torch.sin(pts.extract("x") - self.c * pts.extract("t"))
|
|
sol.labels = self.output_variables
|
|
return sol
|