157 lines
6.3 KiB
Python
Vendored
157 lines
6.3 KiB
Python
Vendored
#!/usr/bin/env python
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# coding: utf-8
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# # Tutorial: The `Equation` Class
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#
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# [](https://colab.research.google.com/github/mathLab/PINA/blob/master/tutorials/tutorial12/tutorial.ipynb)
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# In this tutorial, we will show how to use the `Equation` Class in PINA. Specifically, we will see how use the Class and its inherited classes to enforce residuals minimization in PINNs.
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# # Example: The Burgers 1D equation
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# We will start implementing the viscous Burgers 1D problem Class, described as follows:
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#
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#
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# $$
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# \begin{equation}
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# \begin{cases}
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# \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} &= \nu \frac{\partial^2 u}{ \partial x^2}, \quad x\in(0,1), \quad t>0\\
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# u(x,0) &= -\sin (\pi x)\\
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# u(x,t) &= 0 \quad x = \pm 1\\
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# \end{cases}
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# \end{equation}
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# $$
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#
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# where we set $ \nu = \frac{0.01}{\pi}$.
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#
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# In the class that models this problem we will see in action the `Equation` class and one of its inherited classes, the `FixedValue` class.
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# In[1]:
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## routine needed to run the notebook on Google Colab
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try:
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import google.colab
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IN_COLAB = True
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except:
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IN_COLAB = False
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if IN_COLAB:
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get_ipython().system('pip install "pina-mathlab"')
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#useful imports
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from pina.problem import SpatialProblem, TimeDependentProblem
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from pina.equation import Equation, FixedValue
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from pina.domain import CartesianDomain
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import torch
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from pina.operators import grad, laplacian
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from pina import Condition
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# In[2]:
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class Burgers1D(TimeDependentProblem, SpatialProblem):
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# define the burger equation
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def burger_equation(input_, output_):
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du = grad(output_, input_)
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ddu = grad(du, input_, components=['dudx'])
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return (
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du.extract(['dudt']) +
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output_.extract(['u'])*du.extract(['dudx']) -
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(0.01/torch.pi)*ddu.extract(['ddudxdx'])
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)
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# define initial condition
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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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return output_.extract(['u']) - u_expected
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# assign output/ spatial and temporal variables
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output_variables = ['u']
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spatial_domain = CartesianDomain({'x': [-1, 1]})
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temporal_domain = CartesianDomain({'t': [0, 1]})
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# problem condition statement
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conditions = {
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'bound_cond1': Condition(domain=CartesianDomain({'x': -1, 't': [0, 1]}), equation=FixedValue(0.)),
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'bound_cond2': Condition(domain=CartesianDomain({'x': 1, 't': [0, 1]}), equation=FixedValue(0.)),
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'time_cond': Condition(domain=CartesianDomain({'x': [-1, 1], 't': 0}), equation=Equation(initial_condition)),
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'phys_cond': Condition(domain=CartesianDomain({'x': [-1, 1], 't': [0, 1]}), equation=Equation(burger_equation)),
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}
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#
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# The `Equation` class takes as input a function (in this case it happens twice, with `initial_condition` and `burger_equation`) which computes a residual of an equation, such as a PDE. In a problem class such as the one above, the `Equation` class with such a given input is passed as a parameter in the specified `Condition`.
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#
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# The `FixedValue` class takes as input a value of same dimensions of the output functions; this class can be used to enforce a fixed value for a specific condition, e.g. Dirichlet boundary conditions, as it happens for instance in our example.
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#
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# Once the equations are set as above in the problem conditions, the PINN solver will aim to minimize the residuals described in each equation in the training phase.
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# Available classes of equations include also:
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# - `FixedGradient` and `FixedFlux`: they work analogously to `FixedValue` class, where we can require a constant value to be enforced, respectively, on the gradient of the solution or the divergence of the solution;
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# - `Laplace`: it can be used to enforce the laplacian of the solution to be zero;
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# - `SystemEquation`: we can enforce multiple conditions on the same subdomain through this class, passing a list of residual equations defined in the problem.
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#
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# # Defining a new Equation class
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# `Equation` classes can be also inherited to define a new class. As example, we can see how to rewrite the above problem introducing a new class `Burgers1D`; during the class call, we can pass the viscosity parameter $\nu$:
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# In[3]:
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class Burgers1DEquation(Equation):
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def __init__(self, nu = 0.):
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"""
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Burgers1D class. This class can be
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used to enforce the solution u to solve the viscous Burgers 1D Equation.
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:param torch.float32 nu: the viscosity coefficient. Default value is set to 0.
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"""
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self.nu = nu
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def equation(input_, output_):
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return grad(output_, input_, d='t') +\
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output_*grad(output_, input_, d='x') -\
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self.nu*laplacian(output_, input_, d='x')
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super().__init__(equation)
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# Now we can just pass the above class as input for the last condition, setting $\nu= \frac{0.01}{\pi}$:
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# In[4]:
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class Burgers1D(TimeDependentProblem, SpatialProblem):
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# define initial condition
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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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return output_.extract(['u']) - u_expected
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# assign output/ spatial and temporal variables
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output_variables = ['u']
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spatial_domain = CartesianDomain({'x': [-1, 1]})
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temporal_domain = CartesianDomain({'t': [0, 1]})
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# problem condition statement
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conditions = {
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'bound_cond1': Condition(domain=CartesianDomain({'x': -1, 't': [0, 1]}), equation=FixedValue(0.)),
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'bound_cond2': Condition(domain=CartesianDomain({'x': 1, 't': [0, 1]}), equation=FixedValue(0.)),
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'time_cond': Condition(domain=CartesianDomain({'x': [-1, 1], 't': 0}), equation=Equation(initial_condition)),
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'phys_cond': Condition(domain=CartesianDomain({'x': [-1, 1], 't': [0, 1]}), equation=Burgers1DEquation(0.01/torch.pi)),
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}
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# # What's next?
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# Congratulations on completing the `Equation` class tutorial of **PINA**! As we have seen, you can build new classes that inherit `Equation` to store more complex equations, as the Burgers 1D equation, only requiring to pass the characteristic coefficients of the problem.
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# From now on, you can:
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# - define additional complex equation classes (e.g. `SchrodingerEquation`, `NavierStokeEquation`..)
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# - define more `FixedOperator` (e.g. `FixedCurl`)
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