206 lines
8.8 KiB
ReStructuredText
206 lines
8.8 KiB
ReStructuredText
Tutorial 3: resolution of wave equation with custom Network
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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 the
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``SpatialProblem`` and ``TimeDependentProblem`` class, and the
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``Network`` class for building custom **torch** networks.
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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)\cos(\sqrt{2}\pi), \\\\
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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 nabla, grad
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from pina.model import Network
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from pina import Condition, Span, PINN, 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 = Span({'x': [0, 1], 'y': [0, 1]})
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temporal_domain = Span({'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 = nabla(output_, input_, components=['u'], d=['x', 'y'])
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return nabla_u - u_tt
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def nil_dirichlet(input_, output_):
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value = 0.0
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return output_.extract(['u']) - value
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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(Span({'x': [0, 1], 'y': 1, 't': [0, 1]}), nil_dirichlet),
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'gamma2': Condition(Span({'x': [0, 1], 'y': 0, 't': [0, 1]}), nil_dirichlet),
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'gamma3': Condition(Span({'x': 1, 'y': [0, 1], 't': [0, 1]}), nil_dirichlet),
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'gamma4': Condition(Span({'x': 0, 'y': [0, 1], 't': [0, 1]}), nil_dirichlet),
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't0': Condition(Span({'x': [0, 1], 'y': [0, 1], 't': 0}), initial_condition),
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'D': Condition(Span({'x': [0, 1], 'y': [0, 1], 't': [0, 1]}), 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. With
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the ``Network`` class the users can convert any **torch** model in a
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**PINA** model which uses label tensors with a single line of code. We
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will write a simple residual network using linear layers. Here we
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implement a simple residual network composed by linear torch layers.
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This neural network takes as input the coordinates (in this case
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:math:`x`, :math:`y` and :math:`t`) and provides the unkwown field of
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the Wave problem. The residual of the equations are evaluated at several
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sampling points (which the user can manipulate using the method
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``span_pts``) and the loss minimized by the neural network is the sum of
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the residuals.
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.. code:: ipython3
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class TorchNet(torch.nn.Module):
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def __init__(self):
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super().__init__()
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self.residual = torch.nn.Sequential(torch.nn.Linear(3, 24),
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torch.nn.Tanh(),
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torch.nn.Linear(24, 3))
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self.mlp = torch.nn.Sequential(torch.nn.Linear(3, 64),
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torch.nn.Tanh(),
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torch.nn.Linear(64, 1))
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def forward(self, x):
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residual_x = self.residual(x)
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return self.mlp(x + residual_x)
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# model definition
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model = Network(model = TorchNet(),
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input_variables=problem.input_variables,
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output_variables=problem.output_variables,
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extra_features=None)
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In this tutorial, the neural network is trained for 2000 epochs with a
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learning rate of 0.001. These parameters can be modified as desired. We
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highlight that the generation of the sampling points and the train is
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here encapsulated within the function ``generate_samples_and_train``,
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but only for saving some lines of code in the next cells; that function
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is not mandatory in the **PINA** framework. The training takes
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approximately one minute.
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.. code:: ipython3
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def generate_samples_and_train(model, problem):
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# generate pinn object
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pinn = PINN(problem, model, lr=0.001)
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pinn.span_pts(1000, 'random', locations=['D','t0', 'gamma1', 'gamma2', 'gamma3', 'gamma4'])
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pinn.train(1500, 150)
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return pinn
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pinn = generate_samples_and_train(model, problem)
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.. parsed-literal::
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sum gamma1nil_di gamma2nil_di gamma3nil_di gamma4nil_di t0initial_co Dwave_equati
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[epoch 00000] 4.567502e-01 2.847714e-02 1.962997e-02 9.094939e-03 1.247287e-02 3.838658e-01 3.209481e-03
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sum gamma1nil_di gamma2nil_di gamma3nil_di gamma4nil_di t0initial_co Dwave_equati
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[epoch 00001] 4.184132e-01 1.914901e-02 2.436301e-02 8.384322e-03 1.077990e-02 3.530422e-01 2.694697e-03
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sum gamma1nil_di gamma2nil_di gamma3nil_di gamma4nil_di t0initial_co Dwave_equati
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[epoch 00150] 1.694410e-01 9.840883e-03 1.117415e-02 1.140828e-02 1.003646e-02 1.260622e-01 9.190784e-04
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sum gamma1nil_di gamma2nil_di gamma3nil_di gamma4nil_di t0initial_co Dwave_equati
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[epoch 00300] 1.666860e-01 9.847926e-03 1.122043e-02 1.142906e-02 9.706282e-03 1.237589e-01 7.233715e-04
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sum gamma1nil_di gamma2nil_di gamma3nil_di gamma4nil_di t0initial_co Dwave_equati
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[epoch 00450] 1.564735e-01 8.579318e-03 1.203290e-02 1.264551e-02 8.249855e-03 1.136869e-01 1.279038e-03
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sum gamma1nil_di gamma2nil_di gamma3nil_di gamma4nil_di t0initial_co Dwave_equati
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[epoch 00600] 1.281068e-01 5.976059e-03 1.463099e-02 1.191054e-02 7.087692e-03 8.658079e-02 1.920737e-03
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sum gamma1nil_di gamma2nil_di gamma3nil_di gamma4nil_di t0initial_co Dwave_equati
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[epoch 00750] 7.482838e-02 5.880896e-03 1.912235e-02 5.754319e-03 4.252454e-03 3.697925e-02 2.839110e-03
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sum gamma1nil_di gamma2nil_di gamma3nil_di gamma4nil_di t0initial_co Dwave_equati
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[epoch 00900] 3.109156e-02 2.877797e-03 5.560369e-03 3.611543e-03 3.818088e-03 1.117986e-02 4.043903e-03
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sum gamma1nil_di gamma2nil_di gamma3nil_di gamma4nil_di t0initial_co Dwave_equati
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[epoch 01050] 1.969596e-02 2.598281e-03 3.658714e-03 3.426491e-03 3.696677e-03 4.037755e-03 2.278043e-03
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sum gamma1nil_di gamma2nil_di gamma3nil_di gamma4nil_di t0initial_co Dwave_equati
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[epoch 01200] 1.625224e-02 2.496960e-03 3.069649e-03 3.198287e-03 3.420298e-03 2.728654e-03 1.338392e-03
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sum gamma1nil_di gamma2nil_di gamma3nil_di gamma4nil_di t0initial_co Dwave_equati
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[epoch 01350] 1.430180e-02 2.350929e-03 2.700139e-03 2.961276e-03 3.141905e-03 2.189825e-03 9.577314e-04
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[epoch 01500] 1.293717e-02 2.182199e-03 2.440975e-03 2.706538e-03 2.904802e-03 1.891113e-03 8.115429e-04
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After the training is completed one can now plot some results using the
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``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 = 0.6
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plotter.plot(pinn, fixed_variables={'t': 0.6})
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.. image:: tutorial_files/tutorial_12_0.png
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We can also plot the pinn loss during the training to see the decrease.
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.. code:: ipython3
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import matplotlib.pyplot as plt
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plt.figure(figsize=(16, 6))
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plotter.plot_loss(pinn, label='Loss')
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plt.grid()
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plt.legend()
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plt.show()
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.. image:: tutorial_files/tutorial_14_0.png
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You can now trying improving the training by changing network, optimizer
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and its parameters, changin the sampling points,or adding extra
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features!
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