a9b1bd2826
Tutorial update and small fixes * Tutorials update + Tutorial FNO * Create a metric tracker callback * Update PINN for logging * Update plotter for plotting * Small fix LabelTensor * Small fix FNO --------- Co-authored-by: Dario Coscia <dariocoscia@cli-10-110-13-250.WIFIeduroamSTUD.units.it> Co-authored-by: Dario Coscia <dariocoscia@dhcp-176.eduroam.sissa.it>
191 lines
7.1 KiB
ReStructuredText
191 lines
7.1 KiB
ReStructuredText
Tutorial 3: resolution of wave equation with hard constraint PINNs.
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===================================================================
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The problem solution
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~~~~~~~~~~~~~~~~~~~~
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In this tutorial we present how to solve the wave equation using hard
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constraint PINNs. For doing so we will build a costum torch model and
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pass it to the ``PINN`` solver.
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The problem is written in the following form:
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:raw-latex:`\begin{equation}
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\begin{cases}
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\Delta u(x,y,t) = \frac{\partial^2}{\partial t^2} u(x,y,t) \quad \text{in } D, \\\\
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u(x, y, t=0) = \sin(\pi x)\sin(\pi y), \\\\
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u(x, y, t) = 0 \quad \text{on } \Gamma_1 \cup \Gamma_2 \cup \Gamma_3 \cup \Gamma_4,
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\end{cases}
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\end{equation}`
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where :math:`D` is a square domain :math:`[0,1]^2`, and
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:math:`\Gamma_i`, with :math:`i=1,...,4`, are the boundaries of the
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square, and the velocity in the standard wave equation is fixed to one.
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First of all, some useful imports.
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.. code:: ipython3
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import torch
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from pina.problem import SpatialProblem, TimeDependentProblem
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from pina.operators import laplacian, grad
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from pina.geometry import CartesianDomain
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from pina.solvers import PINN
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from pina.trainer import Trainer
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from pina.equation import Equation
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from pina.equation.equation_factory import FixedValue
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from pina import Condition, Plotter
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Now, the wave problem is written in PINA code as a class, inheriting
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from ``SpatialProblem`` and ``TimeDependentProblem`` since we deal with
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spatial, and time dependent variables. The equations are written as
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``conditions`` that should be satisfied in the corresponding domains.
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``truth_solution`` is the exact solution which will be compared with the
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predicted one.
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.. code:: ipython3
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class Wave(TimeDependentProblem, SpatialProblem):
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output_variables = ['u']
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spatial_domain = CartesianDomain({'x': [0, 1], 'y': [0, 1]})
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temporal_domain = CartesianDomain({'t': [0, 1]})
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def wave_equation(input_, output_):
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u_t = grad(output_, input_, components=['u'], d=['t'])
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u_tt = grad(u_t, input_, components=['dudt'], d=['t'])
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nabla_u = laplacian(output_, input_, components=['u'], d=['x', 'y'])
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return nabla_u - u_tt
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def initial_condition(input_, output_):
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u_expected = (torch.sin(torch.pi*input_.extract(['x'])) *
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torch.sin(torch.pi*input_.extract(['y'])))
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return output_.extract(['u']) - u_expected
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conditions = {
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'gamma1': Condition(location=CartesianDomain({'x': [0, 1], 'y': 1, 't': [0, 1]}), equation=FixedValue(0.)),
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'gamma2': Condition(location=CartesianDomain({'x': [0, 1], 'y': 0, 't': [0, 1]}), equation=FixedValue(0.)),
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'gamma3': Condition(location=CartesianDomain({'x': 1, 'y': [0, 1], 't': [0, 1]}), equation=FixedValue(0.)),
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'gamma4': Condition(location=CartesianDomain({'x': 0, 'y': [0, 1], 't': [0, 1]}), equation=FixedValue(0.)),
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't0': Condition(location=CartesianDomain({'x': [0, 1], 'y': [0, 1], 't': 0}), equation=Equation(initial_condition)),
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'D': Condition(location=CartesianDomain({'x': [0, 1], 'y': [0, 1], 't': [0, 1]}), equation=Equation(wave_equation)),
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}
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def wave_sol(self, pts):
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return (torch.sin(torch.pi*pts.extract(['x'])) *
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torch.sin(torch.pi*pts.extract(['y'])) *
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torch.cos(torch.sqrt(torch.tensor(2.))*torch.pi*pts.extract(['t'])))
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truth_solution = wave_sol
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problem = Wave()
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After the problem, a **torch** model is needed to solve the PINN.
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Usually many models are already implemented in ``PINA``, but the user
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has the possibility to build his/her own model in ``pyTorch``. The hard
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constraint we impose are on the boundary of the spatial domain.
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Specificly our solution is written as:
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.. math:: u_{\rm{pinn}} = xy(1-x)(1-y)\cdot NN(x, y, t),
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where :math:`NN` is the neural net output. This neural network takes as
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input the coordinates (in this case :math:`x`, :math:`y` and :math:`t`)
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and provides the unkwown field of the Wave problem. By construction it
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is zero on the boundaries. The residual of the equations are evaluated
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at several sampling points (which the user can manipulate using the
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method ``discretise_domain``) and the loss minimized by the neural
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network is the sum of the residuals.
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.. code:: ipython3
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class HardMLP(torch.nn.Module):
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def __init__(self, input_dim, output_dim):
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super().__init__()
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self.layers = torch.nn.Sequential(torch.nn.Linear(input_dim, 20),
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torch.nn.Tanh(),
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torch.nn.Linear(20, 20),
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torch.nn.Tanh(),
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torch.nn.Linear(20, output_dim))
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# here in the foward we implement the hard constraints
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def forward(self, x):
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hard = x.extract(['x'])*(1-x.extract(['x']))*x.extract(['y'])*(1-x.extract(['y']))
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return hard*self.layers(x)
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In this tutorial, the neural network is trained for 3000 epochs with a
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learning rate of 0.001 (default in ``PINN``). Training takes
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approximately 1 minute.
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.. code:: ipython3
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pinn = PINN(problem, HardMLP(len(problem.input_variables), len(problem.output_variables)))
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problem.discretise_domain(1000, 'random', locations=['D','t0', 'gamma1', 'gamma2', 'gamma3', 'gamma4'])
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trainer = Trainer(pinn, max_epochs=3000)
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trainer.train()
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.. parsed-literal::
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GPU available: False, used: False
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TPU available: False, using: 0 TPU cores
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IPU available: False, using: 0 IPUs
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HPU available: False, using: 0 HPUs
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| Name | Type | Params
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----------------------------------------
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0 | _loss | MSELoss | 0
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1 | _neural_net | Network | 521
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----------------------------------------
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521 Trainable params
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0 Non-trainable params
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521 Total params
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0.002 Total estimated model params size (MB)
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.. parsed-literal::
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Epoch 2999: : 1it [00:00, 79.33it/s, v_num=5, mean_loss=0.00119, D_loss=0.00542, t0_loss=0.0017, gamma1_loss=0.000, gamma2_loss=0.000, gamma3_loss=0.000, gamma4_loss=0.000]
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.. parsed-literal::
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`Trainer.fit` stopped: `max_epochs=3000` reached.
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.. parsed-literal::
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Epoch 2999: : 1it [00:00, 68.62it/s, v_num=5, mean_loss=0.00119, D_loss=0.00542, t0_loss=0.0017, gamma1_loss=0.000, gamma2_loss=0.000, gamma3_loss=0.000, gamma4_loss=0.000]
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Notice that the loss on the boundaries of the spatial domain is exactly
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zero, as expected! After the training is completed one can now plot some
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results using the ``Plotter`` class of **PINA**.
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.. code:: ipython3
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plotter = Plotter()
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# plotting at fixed time t = 0.0
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plotter.plot(trainer, fixed_variables={'t': 0.0})
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# plotting at fixed time t = 0.5
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plotter.plot(trainer, fixed_variables={'t': 0.5})
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# plotting at fixed time t = 1.
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plotter.plot(trainer, fixed_variables={'t': 1.0})
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.. image:: tutorial_files/tutorial_12_0.png
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.. image:: tutorial_files/tutorial_12_1.png
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.. image:: tutorial_files/tutorial_12_2.png
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