""" Module for CausalPINN """
import torch
from torch.optim.lr_scheduler import ConstantLR
from .pinn import PINN
from pina.problem import TimeDependentProblem
from pina.utils import check_consistency
[docs]
class CausalPINN(PINN):
r"""
Causal Physics Informed Neural Network (PINN) solver class.
This class implements Causal Physics Informed Neural
Network solvers, using a user specified ``model`` to solve a specific
``problem``. It can be used for solving both forward and inverse problems.
The Causal Physics Informed Network aims to find
the solution :math:`\mathbf{u}:\Omega\rightarrow\mathbb{R}^m`
of the differential problem:
.. math::
\begin{cases}
\mathcal{A}[\mathbf{u}](\mathbf{x})=0\quad,\mathbf{x}\in\Omega\\
\mathcal{B}[\mathbf{u}](\mathbf{x})=0\quad,
\mathbf{x}\in\partial\Omega
\end{cases}
minimizing the loss function
.. math::
\mathcal{L}_{\rm{problem}} = \frac{1}{N_t}\sum_{i=1}^{N_t}
\omega_{i}\mathcal{L}_r(t_i),
where:
.. math::
\mathcal{L}_r(t) = \frac{1}{N}\sum_{i=1}^N
\mathcal{L}(\mathcal{A}[\mathbf{u}](\mathbf{x}_i, t)) +
\frac{1}{N}\sum_{i=1}^N
\mathcal{L}(\mathcal{B}[\mathbf{u}](\mathbf{x}_i, t))
and,
.. math::
\omega_i = \exp\left(\epsilon \sum_{k=1}^{i-1}\mathcal{L}_r(t_k)\right).
:math:`\epsilon` is an hyperparameter, default set to :math:`100`, while
:math:`\mathcal{L}` is a specific loss function,
default Mean Square Error:
.. math::
\mathcal{L}(v) = \| v \|^2_2.
.. seealso::
**Original reference**: Wang, Sifan, Shyam Sankaran, and Paris
Perdikaris. "Respecting causality for training physics-informed
neural networks." Computer Methods in Applied Mechanics
and Engineering 421 (2024): 116813.
DOI `10.1016 <https://doi.org/10.1016/j.cma.2024.116813>`_.
.. note::
This class can only work for problems inheriting
from at least
:class:`~pina.problem.timedep_problem.TimeDependentProblem` class.
"""
def __init__(
self,
problem,
model,
extra_features=None,
loss=torch.nn.MSELoss(),
optimizer=torch.optim.Adam,
optimizer_kwargs={"lr": 0.001},
scheduler=ConstantLR,
scheduler_kwargs={"factor": 1, "total_iters": 0},
eps=100,
):
"""
:param AbstractProblem problem: The formulation of the problem.
:param torch.nn.Module model: The neural network model to use.
:param torch.nn.Module loss: The loss function used as minimizer,
default :class:`torch.nn.MSELoss`.
:param torch.nn.Module extra_features: The additional input
features to use as augmented input.
:param torch.optim.Optimizer optimizer: The neural network optimizer to
use; default is :class:`torch.optim.Adam`.
:param dict optimizer_kwargs: Optimizer constructor keyword args.
:param torch.optim.LRScheduler scheduler: Learning
rate scheduler.
:param dict scheduler_kwargs: LR scheduler constructor keyword args.
:param int | float eps: The exponential decay parameter. Note that this
value is kept fixed during the training, but can be changed by means
of a callback, e.g. for annealing.
"""
super().__init__(
problem=problem,
model=model,
extra_features=extra_features,
loss=loss,
optimizer=optimizer,
optimizer_kwargs=optimizer_kwargs,
scheduler=scheduler,
scheduler_kwargs=scheduler_kwargs,
)
# checking consistency
check_consistency(eps, (int, float))
self._eps = eps
if not isinstance(self.problem, TimeDependentProblem):
raise ValueError(
"Casual PINN works only for problems"
"inheritig from TimeDependentProblem."
)
[docs]
def loss_phys(self, samples, equation):
"""
Computes the physics loss for the Causal PINN solver based on given
samples and equation.
:param LabelTensor samples: The samples to evaluate the physics loss.
:param EquationInterface equation: The governing equation
representing the physics.
:return: The physics loss calculated based on given
samples and equation.
:rtype: LabelTensor
"""
# split sequentially ordered time tensors into chunks
chunks, labels = self._split_tensor_into_chunks(samples)
# compute residuals - this correspond to ordered loss functions
# values for each time step. We apply `flatten` such that after
# concataning the residuals we obtain a tensor of shape #chunks
time_loss = []
for chunk in chunks:
chunk.labels = labels
# classical PINN loss
residual = self.compute_residual(samples=chunk, equation=equation)
loss_val = self.loss(
torch.zeros_like(residual, requires_grad=True), residual
)
time_loss.append(loss_val)
# store results
self.store_log(loss_value=float(sum(time_loss) / len(time_loss)))
# concatenate residuals
time_loss = torch.stack(time_loss)
# compute weights (without the gradient storing)
with torch.no_grad():
weights = self._compute_weights(time_loss)
return (weights * time_loss).mean()
@property
def eps(self):
"""
The exponential decay parameter.
"""
return self._eps
@eps.setter
def eps(self, value):
"""
Setter method for the eps parameter.
:param float value: The exponential decay parameter.
"""
check_consistency(value, float)
self._eps = value
def _sort_label_tensor(self, tensor):
"""
Sorts the label tensor based on time variables.
:param LabelTensor tensor: The label tensor to be sorted.
:return: The sorted label tensor based on time variables.
:rtype: LabelTensor
"""
# labels input tensors
labels = tensor.labels
# extract time tensor
time_tensor = tensor.extract(self.problem.temporal_domain.variables)
# sort the time tensors (this is very bad for GPU)
_, idx = torch.sort(time_tensor.tensor.flatten())
tensor = tensor[idx]
tensor.labels = labels
return tensor
def _split_tensor_into_chunks(self, tensor):
"""
Splits the label tensor into chunks based on time.
:param LabelTensor tensor: The label tensor to be split.
:return: Tuple containing the chunks and the original labels.
:rtype: Tuple[List[LabelTensor], List]
"""
# labels input tensors
labels = tensor.labels
# labels input tensors
tensor = self._sort_label_tensor(tensor)
# extract time tensor
time_tensor = tensor.extract(self.problem.temporal_domain.variables)
# count unique tensors in time
_, idx_split = time_tensor.unique(return_counts=True)
# splitting
chunks = torch.split(tensor, tuple(idx_split))
return chunks, labels # return chunks
def _compute_weights(self, loss):
"""
Computes the weights for the physics loss based on the cumulative loss.
:param LabelTensor loss: The physics loss values.
:return: The computed weights for the physics loss.
:rtype: LabelTensor
"""
# compute comulative loss and multiply by epsilos
cumulative_loss = self._eps * torch.cumsum(loss, dim=0)
# return the exponential of the weghited negative cumulative sum
return torch.exp(-cumulative_loss)
