136 lines
5.6 KiB
Python
Vendored
136 lines
5.6 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 hard constraint PINNs.
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# ## The problem definition
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# In this tutorial we present how to solve the wave equation using hard constraint PINNs. For doing so we will build a costum torch model and pass it to the `PINN` solver.
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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[1]:
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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 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[2]:
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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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# ## Hard Constraint Model
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# After the problem, a **torch** model is needed to solve the PINN. Usually, many models are already implemented in `PINA`, but the user has the possibility to build his/her own model in `PyTorch`. The hard constraint we impose is on the boundary of the spatial domain. Specifically, our solution is written as:
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#
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# $$ u_{\rm{pinn}} = xy(1-x)(1-y)\cdot NN(x, y, t), $$
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#
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# where $NN$ is the neural net output. This neural network takes as input the coordinates (in this case $x$, $y$ and $t$) and provides the unknown field $u$. By construction, it is zero on the boundaries. The residuals of the equations are evaluated at several sampling points (which the user can manipulate using the method `discretise_domain`) and the loss minimized by the neural network is the sum of the residuals.
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# In[3]:
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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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# ## Train and Inference
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# In this tutorial, the neural network is trained for 3000 epochs with a learning rate of 0.001 (default in `PINN`). Training takes approximately 1 minute.
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# In[4]:
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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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# Notice that the loss on the boundaries of the spatial domain is exactly zero, as expected! After the training is completed one can now plot some results using the `Plotter` class of **PINA**.
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# In[5]:
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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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