154 lines
5.8 KiB
Python
Vendored
154 lines
5.8 KiB
Python
Vendored
#!/usr/bin/env python
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# coding: utf-8
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# # Tutorial 3: resolution of wave equation with custom Network
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# ### The problem solution
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# In this tutorial we present how to solve the wave equation using the `SpatialProblem` and `TimeDependentProblem` class, and the `Network` class for building custom **torch** networks.
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#
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# The problem is written in the following form:
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#
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# \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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#
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# where $D$ is a square domain $[0,1]^2$, and $\Gamma_i$, with $i=1,...,4$, are the boundaries of the 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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# In[2]:
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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 from `SpatialProblem` and `TimeDependentProblem` since we deal with spatial, and time dependent variables. The equations are written as `conditions` that should be satisfied in the corresponding domains. `truth_solution` is the exact solution which will be compared with the predicted one.
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# In[3]:
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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 the `Network` class the users can convert any **torch** model in a **PINA** model which uses label tensors with a single line of code. We will write a simple residual network using linear layers. Here we implement a simple residual network composed by linear torch layers.
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#
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# This neural network takes as input the coordinates (in this case $x$, $y$ and $t$) and provides the unkwown field of the Wave problem. The residual of the equations are evaluated at several sampling points (which the user can manipulate using the method `span_pts`) and the loss minimized by the neural network is the sum of the residuals.
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# In[4]:
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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 learning rate of 0.001. These parameters can be modified as desired.
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# We highlight that the generation of the sampling points and the train is here encapsulated within the function `generate_samples_and_train`, but only for saving some lines of code in the next cells; that function is not mandatory in the **PINA** framework. The training takes approximately one minute.
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# In[5]:
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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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# After the training is completed one can now plot some results using the `Plotter` class of **PINA**.
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# In[11]:
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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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# We can also plot the pinn loss during the training to see the decrease.
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# In[12]:
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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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# You can now trying improving the training by changing network, optimizer and its parameters, changin the sampling points,or adding extra features!
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