import torch
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class HeatMultiFrequency:
r"""Multi-frequency heat equation, with :math:`0` boundary condition
.. math::
\frac{\partial u }{\partial t} = \Delta u
where :math:`t \in [0,T],\quad(x_1,x_2)\in [-1,1]^2`,
with the boundary condition :math:`u(t, \pm 1, \pm 1) = 0`
Parameters
-----------
mu: torch.Tensor , optional
2D tensor of shape :math:`[N, d]` or 1D tensor of shape :math:`[d]`, where :math:`N` is the number of samples, :math:`d` is the dimension of the frequencies
the coefficient of the heat equation,
if ``None``, it will be randomly generated by :math:`\mu\sim Unif([-1,1]^d)`
d: int, optional
the dimension of the frequencies, if ``mu`` is not ``None``, this parameter will be ignored
if ``mu`` is ``None``, it will be used to generate the random ``mu``
"""
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def __init__(self, mu=None, d=2):
if mu is None:
assert d is not None, "d must be provided if mu is None"
mu = torch.rand(d)
else:
d = mu.shape[-1]
self.mu = mu
self.d = d
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def initial_condition(self, points):
r"""Generate the heat source function at each point in the domain
.. math::
u(0,x_1,x_2,\mu) = -\frac{1}{d}\sum_{m=1}^d \mu_m sin(\pi m x_1)sin(\pi m x_2)/\sqrt m
Parameters
----------
points: torch.Tensor
2D tensor of shape :math:`[|\mathcal V|, 2]`, where :math:`|\mathcal V|` is the number of vertices
all the points must be in :math:`[-1,1]^2`
Returns
-------
torch.Tensor
1D tensor of shape :math:`[|\mathcal V|]` or :math:`[N, |\mathcal V|]`, where :math:`N` is the number of samples, :math:`|\mathcal V|` is the number of vertices
"""
assert points.shape[-1] == 2, f"the shape of points must be [n_points, 2], but got {points.shape}"
assert ((points<=1.0) &(points>=-1)).all(), f"the points must be in [-1,1]^2, but got {points}"
mu = self.mu.to(device=points.device, dtype=points.dtype)
d = self.d
m = torch.arange(1, d+1, dtype=points.dtype, device=points.device)
if len(mu.shape) == 1:
mu = mu[None, :] # (1, d)
m = m[None, :] # (1, d)
x,y = points[:, 0][:, None], points[:, 1][:, None] # (n_points, 1)
else:
mu = mu[:, None, ...] # (N, 1, d)
m = m[None, None, ...] # (1, 1, d)
x, y = points[:, 0][None, :, None], points[:, 1][None, :, None] # (1, n_points, 1)
u0 = - (mu * torch.sin(torch.pi * m * x) * torch.sin(torch.pi * m * y) / torch.sqrt(m) / d).sum(-1)
return u0
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def solution(self, points, t):
r"""Generate the poisson solution function at each point in the domain
.. math::
u(t,x_1,x_2,\mu) = -\frac{1}{d}\sum_{m=1}^d \frac{\mu_m}{\sqrt{m}} e^{-2m^2\pi^2t} sin(\pi m x_1)sin(\pi mx_2)
Parameters
----------
points: torch.Tensor
2D tensor of shape :math:`[|\mathcal V|, 2]`, where :math:`|\mathcal V|` is the number of vertices
t: float
the time
Returns
-------
ut: torch.Tensor
1D tensor of shape :math:`[|\mathcal V|]` or :math:`[N, |\mathcal V|]`, where :math:`N` is the number of samples, :math:`|\mathcal V|` is the number of vertices
"""
assert points.shape[-1] == 2, f"the shape of points must be [n_points, 2], but got {points.shape}"
assert ((points<=1.0) & (points>=-1.0)).all(), f"the points must be in [-1,1]^2, but got {points}"
assert t >= 0, f"t must be non-negative, but got {t}"
mu = self.mu.to(device=points.device, dtype=points.dtype)
d = self.d
m = torch.arange(1, d+1, dtype=points.dtype, device=points.device)
if len(mu.shape) == 1:
mu = mu[None, ...] # (1, d)
m = m[None, ...] # (1, d)
x,y = points[:, 0][:, None], points[:, 1][:, None] # (n_points, 1)
else:
mu = mu[:, None, ...] # (N, 1, d)
m = m[None, None, ...] # (1, 1, d)
x, y = points[:, 0][None, :, None], points[:, 1][None, :, None] # (1, n_points, 1)
ut = - (mu * torch.sin(torch.pi * m * x) * torch.sin(torch.pi * m * y) * torch.exp(-2 * m * m * torch.pi * torch.pi * t)/ torch.sqrt(m) / d).sum(-1)
return ut