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GiovanniCanaliGiovanniCanali
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enhance problem formulation
1 parent 0f36713 commit dcac3a9

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Lines changed: 263 additions & 17 deletions

‎pina/_src/problem/zoo/acoustic_wave_problem.py‎

Lines changed: 44 additions & 2 deletions
Original file line numberDiff line numberDiff line change
@@ -28,8 +28,50 @@ def initial_condition(input_, output_):
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class AcousticWaveProblem(TimeDependentProblem, SpatialProblem):
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r"""
31-
Implementation of the acoustic wave problem in the spatial interval
32-
:math:`[0, 1]` and temporal interval :math:`[0, 1]`.
31+
Implementation of the one-dimensional acoustic wave problem on the
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space-time domain :math:`\Omega\times T = [0, 1] \times [0, 1]`.
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The problem is governed by the acoustic wave equation
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.. math::
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\frac{\partial^2 u}{\partial t^2}
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=
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c^2 \frac{\partial^2 u}{\partial x^2},
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where :math:`u = u(x, t)` is the solution field and :math:`c > 0` is the
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wave propagation speed.
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Homogeneous Dirichlet boundary conditions are imposed at the spatial
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boundaries:
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.. math::
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u(0, t) = u(1, t) = 0, \qquad t \in [0, 1].
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The initial displacement is prescribed as
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.. math::
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u(x, 0) = \sin(\pi x) + \frac{1}{2}\sin(4\pi x),
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\qquad x \in [0, 1],
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together with zero initial velocity:
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.. math::
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\frac{\partial u}{\partial t}(x, 0) = 0,
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\qquad x \in [0, 1].
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The analytical solution is given by
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.. math::
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u(x, t)
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=
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\sin(\pi x)\cos(c\pi t)
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+
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\frac{1}{2}\sin(4\pi x)\cos(4c\pi t).
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.. seealso::
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‎pina/_src/problem/zoo/advection_problem.py‎

Lines changed: 33 additions & 3 deletions
Original file line numberDiff line numberDiff line change
@@ -24,9 +24,39 @@ def initial_condition(input_, output_):
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class AdvectionProblem(SpatialProblem, TimeDependentProblem):
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r"""
27-
Implementation of the advection problem in the spatial interval
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:math:`[0, 2 \pi]` and temporal interval :math:`[0, 1]` with periodic
29-
boundary conditions.
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Implementation of the one-dimensional advection problem on the space-time
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domain :math:`\Omega\times T = [0, 2\pi] \times [0, 1]`.
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The problem is governed by the linear advection equation
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.. math::
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\frac{\partial u}{\partial t}
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+
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c \frac{\partial u}{\partial x}
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=
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0,
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where :math:`u = u(x, t)` is the solution field and :math:`c` is the
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advection velocity.
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Periodic boundary conditions are imposed at the spatial boundaries:
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.. math::
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u(0, t) = u(2\pi, t), \qquad t \in [0, 1].
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The initial condition is prescribed as
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51+
.. math::
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u(x, 0) = \sin(x), \qquad x \in [0, 2\pi].
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The analytical solution is given by
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.. math::
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u(x, t) = \sin(x - ct).
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.. seealso::
3262

‎pina/_src/problem/zoo/allen_cahn_problem.py‎

Lines changed: 29 additions & 3 deletions
Original file line numberDiff line numberDiff line change
@@ -26,9 +26,35 @@ def initial_condition(input_, output_):
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class AllenCahnProblem(TimeDependentProblem, SpatialProblem):
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r"""
29-
Implementation of the Allen Cahn problem in the spatial interval
30-
:math:`[-1, 1]` and temporal interval :math:`[0, 1]` with periodic
31-
boundary conditions.
29+
Implementation of the one-dimensional Allen-Cahn problem on the space-time
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domain :math:`\Omega\times T = [-1, 1] \times [0, 1]`.
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The problem is governed by the Allen-Cahn equation
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.. math::
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\frac{\partial u}{\partial t}
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-
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\alpha \frac{\partial^2 u}{\partial x^2}
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+
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\beta \left(u^3 - u\right)
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=
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0,
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where :math:`u = u(x, t)` is the solution field, :math:`\alpha` is the
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diffusion coefficient, and :math:`\beta` is the reaction coefficient.
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Periodic boundary conditions are imposed at the spatial boundaries:
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.. math::
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u(-1, t) = u(1, t), \qquad t \in [0, 1].
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The initial condition is prescribed as
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.. math::
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u(x, 0) = x^2 \cos(\pi x), \qquad x \in [-1, 1].
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.. seealso::
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‎pina/_src/problem/zoo/diffusion_reaction_problem.py‎

Lines changed: 57 additions & 2 deletions
Original file line numberDiff line numberDiff line change
@@ -35,8 +35,63 @@ def initial_condition(input_, output_):
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class DiffusionReactionProblem(TimeDependentProblem, SpatialProblem):
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r"""
38-
Implementation of the diffusion-reaction problem in the spatial interval
39-
:math:`[-\pi, \pi]` and temporal interval :math:`[0, 1]`.
38+
Implementation of the one-dimensional diffusion-reaction problem on the
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space-time domain :math:`\Omega\times T = [-\pi, \pi] \times [0, 1]`.
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The problem is governed by the forced diffusion-reaction equation
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.. math::
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\frac{\partial u}{\partial t}
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-
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\alpha \frac{\partial^2 u}{\partial x^2}
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=
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f(x, t),
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where :math:`u = u(x, t)` is the solution field, :math:`\alpha` is the
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diffusion coefficient, and :math:`f(x, t)` is a forcing term.
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Homogeneous Dirichlet boundary conditions are imposed at the spatial
55+
boundaries:
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.. math::
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u(-\pi, t) = u(\pi, t) = 0, \qquad t \in [0, 1].
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The initial condition is prescribed as
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.. math::
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65+
u(x, 0)
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=
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\sin(x)
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+
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\frac{1}{2}\sin(2x)
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+
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\frac{1}{3}\sin(3x)
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+
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\frac{1}{4}\sin(4x)
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+
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\frac{1}{8}\sin(8x).
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The analytical solution is given by
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.. math::
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u(x, t)
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=
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e^{-t}
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\left(
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\sin(x)
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+
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\frac{1}{2}\sin(2x)
88+
+
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\frac{1}{3}\sin(3x)
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+
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\frac{1}{4}\sin(4x)
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+
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\frac{1}{8}\sin(8x)
94+
\right).
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.. seealso::
4297

‎pina/_src/problem/zoo/helmholtz_problem.py‎

Lines changed: 39 additions & 2 deletions
Original file line numberDiff line numberDiff line change
@@ -11,8 +11,45 @@
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class HelmholtzProblem(SpatialProblem):
1313
r"""
14-
Implementation of the Helmholtz problem in the square domain
15-
:math:`[-1, 1] \times [-1, 1]`.
14+
Implementation of the two-dimensional Helmholtz problem on the square domain
15+
:math:`\Omega = [-1, 1] \times [-1, 1]`.
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The problem is governed by the forced Helmholtz equation
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.. math::
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\Delta u + k u = f(x, y),
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where :math:`u = u(x, y)` is the solution field, :math:`k` is the squared
24+
wavenumber, and :math:`f(x, y)` is a forcing term.
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26+
Homogeneous Dirichlet boundary conditions are imposed on the boundary of
27+
the domain:
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.. math::
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31+
u(x, y) = 0, \qquad (x, y) \in \partial \Omega.
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The analytical solution is given by
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35+
.. math::
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u(x, y)
38+
=
39+
\sin(\alpha_x \pi x)
40+
\sin(\alpha_y \pi y),
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with forcing term
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44+
.. math::
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f(x, y)
47+
=
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\left[
49+
k - (\alpha_x^2 + \alpha_y^2)\pi^2
50+
\right]
51+
\sin(\alpha_x \pi x)
52+
\sin(\alpha_y \pi y).
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.. seealso::
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‎pina/_src/problem/zoo/inverse_poisson_problem.py‎

Lines changed: 27 additions & 3 deletions
Original file line numberDiff line numberDiff line change
@@ -74,9 +74,33 @@ def laplace_equation(input_, output_, params_):
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class InversePoisson2DSquareProblem(SpatialProblem, InverseProblem):
7676
r"""
77-
Implementation of the inverse 2-dimensional Poisson problem in the square
78-
domain :math:`[0, 1] \times [0, 1]`, with unknown parameter domain
79-
:math:`[-1, 1] \times [-1, 1]`.
77+
Implementation of the inverse two-dimensional Poisson problem on the square
78+
domain :math:`\Omega = [-2, 2] \times [-2, 2]`, with unknown parameter
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domain :math:`\Theta = [-1, 1] \times [-1, 1]`.
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The problem is governed by the parameterized Poisson equation
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.. math::
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85+
\Delta u
86+
=
87+
\exp\left(
88+
-2(x - \mu_1)^2
89+
-2(y - \mu_2)^2
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\right),
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92+
where :math:`u = u(x, y)` is the solution field and :math:`\mu_1, \mu_2` are
93+
unknown parameters controlling the forcing term.
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95+
Homogeneous Dirichlet boundary conditions are imposed on the boundary of the
96+
domain:
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98+
.. math::
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100+
u(x, y) = 0, \qquad (x, y) \in \partial \Omega.
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The inverse problem aims to infer the unknown parameters :math:`\mu_1` and
103+
:math:`\mu_2` from solution data.
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The `"data"` condition is added only if the required files are downloaded
82106
successfully.

‎pina/_src/problem/zoo/poisson_problem.py‎

Lines changed: 34 additions & 2 deletions
Original file line numberDiff line numberDiff line change
@@ -26,8 +26,40 @@ def forcing_term(input_):
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2727
class Poisson2DSquareProblem(SpatialProblem):
2828
r"""
29-
Implementation of the 2-dimensional Poisson problem in the square domain
30-
:math:`[0, 1] \times [0, 1]`.
29+
Implementation of the two-dimensional Poisson problem on the square domain
30+
:math:`\Omega = [0, 1] \times [0, 1]`.
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32+
The problem is governed by the Poisson equation
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34+
.. math::
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36+
\Delta u = f(x, y),
37+
38+
where :math:`u = u(x, y)` is the solution field and :math:`f(x, y)` is the
39+
forcing term.
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41+
Homogeneous Dirichlet boundary conditions are imposed on the boundary of the
42+
domain:
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44+
.. math::
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46+
u(x, y) = 0, \qquad (x, y) \in \partial \Omega.
47+
48+
The forcing term is given by
49+
50+
.. math::
51+
52+
f(x, y)
53+
=
54+
2\pi^2 \sin(\pi x)\sin(\pi y).
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56+
The analytical solution is given by
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58+
.. math::
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60+
u(x, y)
61+
=
62+
-\sin(\pi x)\sin(\pi y).
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:Example:
3365

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