"""
Module for operators vectorize implementation. Differential operators are used to write any differential problem.
These operators are implemented to work on different accellerators: CPU, GPU, TPU or MPS.
All operators take as input a tensor onto which computing the operator, a tensor with respect
to which computing the operator, the name of the output variables to calculate the operator
for (in case of multidimensional functions), and the variables name on which the operator is calculated.
"""
import torch
from pina.label_tensor import LabelTensor
[docs]
def grad(output_, input_, components=None, d=None):
"""
Perform gradient operation. The operator works for vectorial and scalar
functions, with multiple input coordinates.
:param LabelTensor output_: the output tensor onto which computing the
gradient.
:param LabelTensor input_: the input tensor with respect to which computing
the gradient.
:param list(str) components: the name of the output variables to calculate
the gradient for. It should be a subset of the output labels. If None,
all the output variables are considered. Default is None.
:param list(str) d: the name of the input variables on which the gradient is
calculated. d should be a subset of the input labels. If None, all the
input variables are considered. Default is None.
:return: the gradient tensor.
:rtype: LabelTensor
"""
def grad_scalar_output(output_, input_, d):
"""
Perform gradient operation for a scalar output.
:param LabelTensor output_: the output tensor onto which computing the
gradient. It has to be a column tensor.
:param LabelTensor input_: the input tensor with respect to which
computing the gradient.
:param list(str) d: the name of the input variables on which the
gradient is calculated. d should be a subset of the input labels. If
None, all the input variables are considered. Default is None.
:raises RuntimeError: a vectorial function is passed.
:raises RuntimeError: missing derivative labels.
:return: the gradient tensor.
:rtype: LabelTensor
"""
if len(output_.labels) != 1:
raise RuntimeError("only scalar function can be differentiated")
if not all([di in input_.labels for di in d]):
raise RuntimeError("derivative labels missing from input tensor")
output_fieldname = output_.labels[0]
gradients = torch.autograd.grad(
output_,
input_,
grad_outputs=torch.ones(
output_.size(), dtype=output_.dtype, device=output_.device
),
create_graph=True,
retain_graph=True,
allow_unused=True,
)[0]
gradients.labels = input_.labels
gradients = gradients.extract(d)
gradients.labels = [f"d{output_fieldname}d{i}" for i in d]
return gradients
if not isinstance(input_, LabelTensor):
raise TypeError
if d is None:
d = input_.labels
if components is None:
components = output_.labels
if output_.shape[1] == 1: # scalar output ################################
if components != output_.labels:
raise RuntimeError
gradients = grad_scalar_output(output_, input_, d)
elif output_.shape[1] >= 2: # vector output ##############################
for i, c in enumerate(components):
c_output = output_.extract([c])
if i == 0:
gradients = grad_scalar_output(c_output, input_, d)
else:
gradients = gradients.append(
grad_scalar_output(c_output, input_, d)
)
else:
raise NotImplementedError
return gradients
[docs]
def div(output_, input_, components=None, d=None):
"""
Perform divergence operation. The operator works for vectorial functions,
with multiple input coordinates.
:param LabelTensor output_: the output tensor onto which computing the
divergence.
:param LabelTensor input_: the input tensor with respect to which computing
the divergence.
:param list(str) components: the name of the output variables to calculate
the divergence for. It should be a subset of the output labels. If None,
all the output variables are considered. Default is None.
:param list(str) d: the name of the input variables on which the divergence
is calculated. d should be a subset of the input labels. If None, all
the input variables are considered. Default is None.
:raises TypeError: div operator works only for LabelTensor.
:raises ValueError: div operator works only for vector fields.
:raises ValueError: div operator must derive all components with
respect to all coordinates.
:return: the divergenge tensor.
:rtype: LabelTensor
"""
if not isinstance(input_, LabelTensor):
raise TypeError
if d is None:
d = input_.labels
if components is None:
components = output_.labels
if output_.shape[1] < 2 or len(components) < 2:
raise ValueError("div supported only for vector fields")
if len(components) != len(d):
raise ValueError
grad_output = grad(output_, input_, components, d)
div = torch.zeros(input_.shape[0], 1, device=output_.device)
labels = [None] * len(components)
for i, (c, d) in enumerate(zip(components, d)):
c_fields = f"d{c}d{d}"
div[:, 0] += grad_output.extract(c_fields).sum(axis=1)
labels[i] = c_fields
div = div.as_subclass(LabelTensor)
div.labels = ["+".join(labels)]
return div
[docs]
def laplacian(output_, input_, components=None, d=None, method="std"):
"""
Compute Laplace operator. The operator works for vectorial and
scalar functions, with multiple input coordinates.
:param LabelTensor output_: the output tensor onto which computing the
Laplacian.
:param LabelTensor input_: the input tensor with respect to which computing
the Laplacian.
:param list(str) components: the name of the output variables to calculate
the Laplacian for. It should be a subset of the output labels. If None,
all the output variables are considered. Default is None.
:param list(str) d: the name of the input variables on which the Laplacian
is calculated. d should be a subset of the input labels. If None, all
the input variables are considered. Default is None.
:param str method: used method to calculate Laplacian, defaults to 'std'.
:raises NotImplementedError: 'divgrad' not implemented as method.
:return: The tensor containing the result of the Laplacian operator.
:rtype: LabelTensor
"""
def scalar_laplace(output_, input_, components, d):
"""
Compute Laplace operator for a scalar output.
:param LabelTensor output_: the output tensor onto which computing the
Laplacian. It has to be a column tensor.
:param LabelTensor input_: the input tensor with respect to which
computing the Laplacian.
:param list(str) components: the name of the output variables to
calculate the Laplacian for. It should be a subset of the output
labels. If None, all the output variables are considered.
:param list(str) d: the name of the input variables on which the
Laplacian is computed. d should be a subset of the input labels.
If None, all the input variables are considered. Default is None.
:return: The tensor containing the result of the Laplacian operator.
:rtype: LabelTensor
"""
grad_output = grad(output_, input_, components=components, d=d)
result = torch.zeros(output_.shape[0], 1, device=output_.device)
for i, label in enumerate(grad_output.labels):
gg = grad(grad_output, input_, d=d, components=[label])
result[:, 0] += super(torch.Tensor, gg.T).__getitem__(i)
return result
if d is None:
d = input_.labels
if components is None:
components = output_.labels
if method == "divgrad":
raise NotImplementedError("divgrad not implemented as method")
# TODO fix
# grad_output = grad(output_, input_, components, d)
# result = div(grad_output, input_, d=d)
elif method == "std":
if len(components) == 1:
result = scalar_laplace(output_, input_, components, d)
labels = [f"dd{components[0]}"]
else:
result = torch.empty(
size=(input_.shape[0], len(components)),
dtype=output_.dtype,
device=output_.device,
)
labels = [None] * len(components)
for idx, c in enumerate(components):
result[:, idx] = scalar_laplace(output_, input_, c, d).flatten()
labels[idx] = f"dd{c}"
result = result.as_subclass(LabelTensor)
result.labels = labels
return result
[docs]
def advection(output_, input_, velocity_field, components=None, d=None):
"""
Perform advection operation. The operator works for vectorial functions,
with multiple input coordinates.
:param LabelTensor output_: the output tensor onto which computing the
advection.
:param LabelTensor input_: the input tensor with respect to which computing
the advection.
:param str velocity_field: the name of the output variables which is used
as velocity field. It should be a subset of the output labels.
:param list(str) components: the name of the output variables to calculate
the Laplacian for. It should be a subset of the output labels. If None,
all the output variables are considered. Default is None.
:param list(str) d: the name of the input variables on which the advection
is calculated. d should be a subset of the input labels. If None, all
the input variables are considered. Default is None.
:return: the tensor containing the result of the advection operator.
:rtype: LabelTensor
"""
if d is None:
d = input_.labels
if components is None:
components = output_.labels
tmp = (
grad(output_, input_, components, d)
.reshape(-1, len(components), len(d))
.transpose(0, 1)
)
tmp *= output_.extract(velocity_field)
return tmp.sum(dim=2).T
