tutorial validation (#185)
Co-authored-by: Ben Volokh <89551265+benv123@users.noreply.github.com>
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Nicola Demo
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tutorials/tutorial1/tutorial.ipynb
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tutorials/tutorial1/tutorial.py
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tutorials/tutorial1/tutorial.py
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@@ -3,18 +3,19 @@
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# # Tutorial 1: Physics Informed Neural Networks on PINA
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# In this tutorial we will show the typical use case of PINA on a toy problem solved by Physics Informed Problems. Specifically, the tutorial aims to introduce the following topics:
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# In this tutorial, we will demonstrate a typical use case of PINA on a toy problem. Specifically, the tutorial aims to introduce the following topics:
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#
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# * Defining a PINA Problem,
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# * Build a `PINN` Solver,
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# * Building a `pinn` object,
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# * Sampling points in a domain
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#
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# We will show in detailed each step, and at the end we will solve a very simple problem with PINA.
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# These are the three main steps needed **before** training a Physics Informed Neural Network (PINN). We will show each step in detail, and at the end, we will solve the problem.
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# ## Defining a Problem
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# ## PINA Problem
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# ### Initialize the Problem class
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# The problem definition in the PINA framework is done by building a phython `class`, inherited from `AbsractProblem`. A problem is an object which explains what the solver is supposed to solve. For Physics Informed Neural Networks, a problem can be inherited from one or more problem (already implemented) classes (`SpatialProblem`, `TimeDependentProblem`, `ParametricProblem`), depending on the nature of the problem treated.
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# Let's see an example to better understand:
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# ### Initialize the `Problem` class
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# Problem definition in the PINA framework is done by building a python `class`, which inherits from one or more problem classes (`SpatialProblem`, `TimeDependentProblem`, `ParametricProblem`) depending on the nature of the problem. Below is an example:
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# #### Simple Ordinary Differential Equation
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# Consider the following:
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#
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@@ -27,55 +28,58 @@
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# \end{equation}
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# $$
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#
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# with analytical solution $u(x) = e^x$. In this case we have that our ODE depends only on the spatial variable $x\in(0,1)$ , this means that our problem class is going to be inherited from `SpatialProblem` class:
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# with the analytical solution $u(x) = e^x$. In this case, our ODE depends only on the spatial variable $x\in(0,1)$ , meaning that our `Problem` class is going to be inherited from the `SpatialProblem` class:
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#
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# ```python
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# from pina.problem import SpatialProblem
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# from pina.geometry import CartesianDomain
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# from pina import CartesianProblem
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#
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# class SimpleODE(SpatialProblem):
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#
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# output_variables = ['u']
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# spatial_domain = CartesianDomain({'x': [0, 1]})
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# spatial_domain = CartesianProblem({'x': [0, 1]})
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#
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# # other stuff ...
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# ```
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#
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# Notice that we define `output_variables` as a list of symbols, indicating the output variables of our equation (in this case only $u$). The `spatial_domain` variable indicates where the sample points are going to be sampled in the domain, in this case $x\in(0,1)$
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#
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# What about if we also have a time depencency in the equation? Well in that case our `class` will inherit from both `SpatialProblem` and `TimeDependentProblem`:
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# ```python
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# from pina.problem import SpatialProblem, TimeDependentProblem
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# from pina.geometry import CartesianDomain
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#
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# class TimeSpaceODE(SpatialProblem, TimeDependentProblem):
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#
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# output_variables = ['u']
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# spatial_domain = CartesianDomain({'x': [0, 1]})
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# temporal_domain = CartesianDomain({'x': [0, 1]})
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#
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# # other stuff ...
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# ```
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# where we have included the `temporal_domain` variable indicating the time domain where we want the solution.
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#
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# Summarizing, in PINA we can initialize a problem with a class which is inherited from three base classes: `SpatialProblem`, `TimeDependentProblem`, `ParametricProblem`, depending on the type of problem we are considering. For reference:
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# * `SpatialProblem` $\rightarrow$ spatial variable(s) presented in the differential equation
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# * `TimeDependentProblem` $\rightarrow$ time variable(s) presented in the differential equation
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# * `ParametricProblem` $\rightarrow$ parameter(s) presented in the differential equation
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# Notice that we define `output_variables` as a list of symbols, indicating the output variables of our equation (in this case only $u$). The `spatial_domain` variable indicates where the sample points are going to be sampled in the domain, in this case $x\in[0,1]$.
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# What about if our equation is also time dependent? In this case, our `class` will inherit from both `SpatialProblem` and `TimeDependentProblem`:
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#
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# ### Write the problem class
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# In[1]:
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from pina.problem import SpatialProblem, TimeDependentProblem
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from pina import CartesianDomain
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class TimeSpaceODE(SpatialProblem, TimeDependentProblem):
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output_variables = ['u']
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spatial_domain = CartesianDomain({'x': [0, 1]})
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temporal_domain = CartesianDomain({'t': [0, 1]})
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# other stuff ...
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# where we have included the `temporal_domain` variable, indicating the time domain wanted for the solution.
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#
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# Once the problem class is initialized we need to write the differential equation in PINA language. For doing this we need to load the pina operators found in `pina.operators` module. Let's again consider the Equation (1) and try to write the PINA model class:
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# In summary, using PINA, we can initialize a problem with a class which inherits from three base classes: `SpatialProblem`, `TimeDependentProblem`, `ParametricProblem`, depending on the type of problem we are considering. For reference:
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# * `SpatialProblem` $\rightarrow$ a differential equation with spatial variable(s)
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# * `TimeDependentProblem` $\rightarrow$ a time-dependent differential equation
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# * `ParametricProblem` $\rightarrow$ a parametrized differential equation
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# ### Write the `Problem` class
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#
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# Once the `Problem` class is initialized, we need to represent the differential equation in PINA. In order to do this, we need to load the PINA operators from `pina.operators` module. Again, we'll consider Equation (1) and represent it in PINA:
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# In[2]:
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from pina.problem import SpatialProblem
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from pina.operators import grad
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from pina.geometry import CartesianDomain
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from pina.equation import Equation
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from pina import Condition
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from pina import Condition, CartesianDomain
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from pina.equation.equation import Equation
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import torch
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@@ -91,50 +95,54 @@ class SimpleODE(SpatialProblem):
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# computing the derivative
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u_x = grad(output_, input_, components=['u'], d=['x'])
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# extracting u input variable
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# extracting the u input variable
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u = output_.extract(['u'])
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# calculate residual and return it
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# calculate the residual and return it
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return u_x - u
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# defining initial condition
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# defining the initial condition
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def initial_condition(input_, output_):
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# setting initial value
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# setting the initial value
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value = 1.0
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# extracting u input variable
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# extracting the u input variable
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u = output_.extract(['u'])
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# calculate residual and return it
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# calculate the residual and return it
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return u - value
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# Conditions to hold
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# conditions to hold
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conditions = {
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'x0': Condition(location=CartesianDomain({'x': 0.}), equation=Equation(initial_condition)),
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'D': Condition(location=CartesianDomain({'x': [0, 1]}), equation=Equation(ode_equation)),
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}
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# defining true solution
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# sampled points (see below)
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input_pts = None
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# defining the true solution
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def truth_solution(self, pts):
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return torch.exp(pts.extract(['x']))
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# After the defition of the Class we need to write different class methods, where each method is a function returning a residual. This functions are the ones minimized during the PINN optimization, for the different conditions. For example, in the domain $(0,1)$ the ODE equation (`ode_equation`) must be satisfied, so we write it by putting all the ODE equation on the right hand side, such that we return the zero residual. This is done for all the conditions (`ode_equation`, `initial_condition`). Notice that we do not pass directly a `python` function, but an `Equation` object, which is initialized with the `python` function. This is done so that all the computations, and internal checks are done inside PINA.
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# After we define the `Problem` class, we need to write different class methods, where each method is a function returning a residual. These functions are the ones minimized during PINN optimization, given the initial conditions. For example, in the domain $[0,1]$, the ODE equation (`ode_equation`) must be satisfied. We represent this by returning the difference between subtracting the variable `u` from its gradient (the residual), which we hope to minimize to 0. This is done for all conditions (`ode_equation`, `initial_condition`).
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#
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# Once we have defined the function we need to tell the network where these methods have to be applied. For doing this we use the class `Condition`. In `Condition` we pass the location points and the function to be minimized on those points (other possibilities are allowed, see the documentation for reference).
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# Once we have defined the function, we need to tell the neural network where these methods are to be applied. To do so, we use the `Condition` class. In the `Condition` class, we pass the location points and the function we want minimized on those points (other possibilities are allowed, see the documentation for reference) as parameters.
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#
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# Finally, it's possible to define a `truth_solution` function, which can be useful if we want to plot the results and see how the real solution compares to the expected (true) solution. Notice that the `truth_solution` function is a method of the `PINN` class, but is not mandatory for problem definition.
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#
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# Finally, it's possible to defing the `truth_solution` function, which can be useful if we want to plot the results and see a comparison of real vs expected solution. Notice that `truth_solution` function is a method of the `PINN` class, but it is not mandatory for the problem definition.
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# ## Build PINN object
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# ## Build the `PINN` object
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# In PINA we have already developed different solvers, one of them is `PINN`. The basics requirements for building a `PINN` model are a problem and a model. We have already covered the problem definition. For the model one can use the default models provided in PINA or use a custom model. We will not go into the details of model definition, Tutorial2 and Tutorial3 treat the topic in detail.
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# The basic requirements for building a `PINN` model are a `Problem` and a model. We have just covered the `Problem` definition. For the model parameter, one can use either the default models provided in PINA or a custom model. We will not go into the details of model definition (see Tutorial2 and Tutorial3 for more details on model definition).
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# In[3]:
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from pina.model import FeedForward
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from pina.solvers import PINN
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from pina import PINN
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# initialize the problem
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problem = SimpleODE()
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input_dimensions=len(problem.input_variables)
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)
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# create the PINN object, see the PINN documentation for extra argument in the constructor
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# create the PINN object
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pinn = PINN(problem, model)
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# Creating the pinn object is fairly simple by using the `PINN` class, different optional inputs can be passed: optimizer, batch size, ... (see [documentation](https://mathlab.github.io/PINA/) for reference).
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# Creating the `PINN` object is fairly simple. Different optional parameters include: optimizer, batch size, ... (see [documentation](https://mathlab.github.io/PINA/) for reference).
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# ## Sample points in the domain and create the Trainer
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# ## Sample points in the domain
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# Once the `PINN` object is created, we need to generate the points for starting the optimization. For doing this we use the `.discretise_domain` method of the `AbstractProblem` class. Let's see some methods to sample in $(0,1 )$.
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# Once the `PINN` object is created, we need to generate the points for starting the optimization. To do so, we use the `sample` method of the `CartesianDomain` class. Below are three examples of sampling methods on the $[0,1]$ domain:
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# In[4]:
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# sampling 20 points in (0, 1) with discrite step
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problem.discretise_domain(20, 'grid', locations=['D'])
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# sampling 20 points in [0, 1] through discretization
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pinn.problem.discretise_domain(n=20, mode='grid', variables=['x'])
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# sampling 20 points in (0, 1) with latin hypercube
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problem.discretise_domain(20, 'latin', locations=['D'])
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# sampling 20 points in (0, 1) through latin hypercube samping
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pinn.problem.discretise_domain(n=20, mode='latin', variables=['x'])
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# sampling 20 points in (0, 1) randomly
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problem.discretise_domain(20, 'random', locations=['D'])
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# We are going to use equispaced points for sampling. We need to sample in all the conditions domains. In our case we sample in `D` and `x0`.
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# In[5]:
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# sampling for training
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problem.discretise_domain(1, 'random', locations=['x0'])
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problem.discretise_domain(20, 'grid', locations=['D'])
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pinn.problem.discretise_domain(n=20, mode='random', variables=['x'])
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# ### Very simple training and plotting
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#
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# Once we have defined the PINA model, created a network and sampled points in the domain, we have everything that is necessary for training a `PINN`. For training we use the `Trainer` class. Here we show a very short training and some method for plotting the results. Notice that by default all relevant metrics (e.g. MSE error during training) is going to be tracked using a `lightining` logger, by default `CSVLogger`. If you want to track the metric by yourself without a logger, use `pina.callbacks.MetricTracker`.
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# Once we have defined the PINA model, created a network, and sampled points in the domain, we have everything necessary for training a PINN. To do so, we make use of the `Trainer` class.
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# In[6]:
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# In[5]:
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# create the trainer
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from pina.trainer import Trainer
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from pina.callbacks import MetricTracker
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from pina import Trainer
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trainer = Trainer(solver=pinn, max_epochs=3000, callbacks=[MetricTracker()])
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# initialize trainer
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trainer = Trainer(pinn)
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# train
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# train the model
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trainer.train()
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# After the training we can inspect trainer logged metrics (by default PINA logs mean square error residual loss). The logged metrics can be accessed online using one of the `Lightinig` loggers. The final loss can be accessed by `trainer.logged_metrics`.
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# In[7]:
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# inspecting final loss
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trainer.logged_metrics
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# By using the `Plotter` class from PINA we can also do some quatitative plots of the solution.
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# In[8]:
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from pina.plotter import Plotter
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# plotting the loss
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plotter = Plotter()
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plotter.plot(trainer=trainer)
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# The solution is completely overlapped with the actual one. We can also plot easily the loss:
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# In[9]:
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plotter.plot_loss(trainer=trainer, metric='mean_loss', log_scale=True)
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