Equation Class Tutorial (#287)
* Tutorial Equation 12 * .rst and readme fix for tutorial12 * small fix * modified rst --------- Co-authored-by: Dario Coscia <dariocoscia@Dario-Coscia.local>
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tutorials/tutorial12/tutorial.ipynb
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Tutorial: The `Equation` Class"
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]
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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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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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Example: The Burgers 1D equation"
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"We will start implementing the viscous Burgers 1D problem Class, described as follows:\n",
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"\n",
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"\n",
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"$$\n",
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"\\begin{equation}\n",
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"\\begin{cases}\n",
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"\\frac{\\partial u}{\\partial t} + u \\frac{\\partial u}{\\partial x} &= \\nu \\frac{\\partial^2 u}{ \\partial x^2}, \\quad x\\in(0,1), \\quad t>0\\\\\n",
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"u(x,0) &= -\\sin (\\pi x)\\\\\n",
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"u(x,t) &= 0 \\quad x = \\pm 1\\\\\n",
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"\\end{cases}\n",
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"\\end{equation}\n",
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"$$\n",
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"\n",
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"where we set $ \\nu = \\frac{0.01}{\\pi}$.\n",
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"\n",
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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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]
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {},
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"outputs": [],
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"source": [
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"#useful imports\n",
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"from pina.problem import SpatialProblem, TimeDependentProblem\n",
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"from pina.equation import Equation, FixedValue, FixedGradient, FixedFlux\n",
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"from pina.geometry import CartesianDomain\n",
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"import torch\n",
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"from pina.operators import grad, laplacian\n",
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"from pina import Condition\n",
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"\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [],
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"source": [
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"class Burgers1D(TimeDependentProblem, SpatialProblem):\n",
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"\n",
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" # define the burger equation\n",
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" def burger_equation(input_, output_):\n",
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" du = grad(output_, input_)\n",
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" ddu = grad(du, input_, components=['dudx'])\n",
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" return (\n",
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" du.extract(['dudt']) +\n",
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" output_.extract(['u'])*du.extract(['dudx']) -\n",
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" (0.01/torch.pi)*ddu.extract(['ddudxdx'])\n",
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" )\n",
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"\n",
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" # define initial condition\n",
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" def initial_condition(input_, output_):\n",
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" u_expected = -torch.sin(torch.pi*input_.extract(['x']))\n",
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" return output_.extract(['u']) - u_expected\n",
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"\n",
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" # assign output/ spatial and temporal variables\n",
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" output_variables = ['u']\n",
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" spatial_domain = CartesianDomain({'x': [-1, 1]})\n",
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" temporal_domain = CartesianDomain({'t': [0, 1]})\n",
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"\n",
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" # problem condition statement\n",
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" conditions = {\n",
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" 'gamma1': Condition(location=CartesianDomain({'x': -1, 't': [0, 1]}), equation=FixedValue(0.)),\n",
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" 'gamma2': Condition(location=CartesianDomain({'x': 1, 't': [0, 1]}), equation=FixedValue(0.)),\n",
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" 't0': Condition(location=CartesianDomain({'x': [-1, 1], 't': 0}), equation=Equation(initial_condition)),\n",
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" 'D': Condition(location=CartesianDomain({'x': [-1, 1], 't': [0, 1]}), equation=Equation(burger_equation)),\n",
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" }"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"\n",
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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`. \n",
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"\n",
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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 enforced a fixed value for a specific condition, e.g. Dirichlet boundary conditions, as it happens for instance in our example.\n",
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"\n",
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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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]
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Available classes of equations include also:\n",
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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;\n",
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"- `Laplace`: it can be used to enforce the laplacian of the solution to be zero;\n",
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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.\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Defining a new Equation class"
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]
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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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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]
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{
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"cell_type": "code",
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"execution_count": 13,
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"metadata": {},
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"outputs": [],
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"source": [
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"class Burgers1DEquation(Equation):\n",
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" \n",
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" def __init__(self, nu = 0.):\n",
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" \"\"\"\n",
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" Burgers1D class. This class can be\n",
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" used to enforce the solution u to solve the viscous Burgers 1D Equation.\n",
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" \n",
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" :param torch.float32 nu: the viscosity coefficient. Default value is set to 0.\n",
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" \"\"\"\n",
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" self.nu = nu \n",
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" \n",
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" def equation(input_, output_):\n",
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" return grad(output_, input_, d='x') +\\\n",
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" output_*grad(output_, input_, d='t') -\\\n",
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" self.nu*laplacian(output_, input_, d='x')\n",
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"\n",
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" \n",
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" super().__init__(equation)"
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]
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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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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]
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"metadata": {},
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"outputs": [],
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"source": [
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"class Burgers1D(TimeDependentProblem, SpatialProblem):\n",
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"\n",
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" # define initial condition\n",
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" def initial_condition(input_, output_):\n",
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" u_expected = -torch.sin(torch.pi*input_.extract(['x']))\n",
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" return output_.extract(['u']) - u_expected\n",
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"\n",
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" # assign output/ spatial and temporal variables\n",
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" output_variables = ['u']\n",
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" spatial_domain = CartesianDomain({'x': [-1, 1]})\n",
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" temporal_domain = CartesianDomain({'t': [0, 1]})\n",
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"\n",
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" # problem condition statement\n",
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" conditions = {\n",
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" 'gamma1': Condition(location=CartesianDomain({'x': -1, 't': [0, 1]}), equation=FixedValue(0.)),\n",
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" 'gamma2': Condition(location=CartesianDomain({'x': 1, 't': [0, 1]}), equation=FixedValue(0.)),\n",
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" 't0': Condition(location=CartesianDomain({'x': [-1, 1], 't': 0}), equation=Equation(initial_condition)),\n",
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" 'D': Condition(location=CartesianDomain({'x': [-1, 1], 't': [0, 1]}), equation=Burgers1DEquation(0.01/torch.pi)),\n",
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" }"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# What's next?"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Congratulations on completing the `Equation` class tutorial of **PINA**! As we have seen, you can build new classes that inherits `Equation` to store more complex equations, as the Burgers 1D equation, only requiring to pass the characteristic coefficients of the problem. \n",
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"From now on, you can:\n",
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"- define additional complex equation classes (e.g. `SchrodingerEquation`, `NavierStokeEquation`..)\n",
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"- define more `FixedOperator` (e.g. `FixedCurl`)"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "pina",
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"language": "python",
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"name": "ipython",
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"file_extension": ".py",
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