Add PI-DeepONet example for 1D Poisson equation (#1311)

Author: Samuel Burbulla

docs/demos/operator/poisson_1d_pideeponet.png

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+Poisson equation in 1D
+======================
+
+Problem setup
+-------------
+
+We will learn the solution operator
+
+.. math:: G: f \mapsto u
+
+for the one-dimensional Poisson problem
+
+.. math:: u''(x) = f(x), \qquad x \in [0, 1],
+
+with zero Dirichlet boundary conditions :math:`u(0) = u(1) = 0`.
+
+The source term :math:`f` is supposed to be an arbitrary continuous function.
+
+
+Implementation
+--------------
+
+The solution operator can be learned by training a physics-informed DeepONet.
+
+First, we define the PDE with boundary conditions and the domain:
+
+.. code-block:: python
+
+    def equation(x, y, f):
+        dy_xx = dde.grad.hessian(y, x)
+        return -dy_xx - f
+
+    geom = dde.geometry.Interval(0, 1)
+
+    def u_boundary(_):
+        return 0
+
+    def boundary(_, on_boundary):
+        return on_boundary
+
+    bc = dde.icbc.DirichletBC(geom, u_boundary, boundary)
+
+    pde = dde.data.PDE(geom, equation, bc, num_domain=100, num_boundary=2)
+
+
+Next, we specify the function space for :math:`f` and the corresponding evaluation points.
+For this example, we use the ``dde.data.PowerSeries`` to get the function space
+of polynomials of degree three.
+Together with the PDE, the function space is used to define a
+PDEOperator ``dde.data.PDEOperatorCartesianProd`` that incorporates the PDE into
+the loss function.
+
+.. code-block:: python
+
+    degree = 3
+    space = dde.data.PowerSeries(N=degree + 1)
+
+    num_eval_points = 10
+    evaluation_points = geom.uniform_points(num_eval_points, boundary=True)
+
+    pde_op = dde.data.PDEOperatorCartesianProd(
+        pde,
+        space,
+        evaluation_points,
+        num_function=100,
+    )
+
+
+The DeepONet can be defined using ``dde.nn.DeepONetCartesianProd``.
+The branch net is chosen as a fully connected neural network of size ``[m, 32, p]`` where ``p=32``
+and the trunk net is a fully connected neural network of size ``[dim_x, 32, p]``.
+
+.. code-block:: python
+
+    dim_x = 1
+    p = 32
+    net = dde.nn.DeepONetCartesianProd(
+        [num_eval_points, 32, p],
+        [dim_x, 32, p],
+        activation="tanh",
+        kernel_initializer="Glorot normal",
+    )
+
+
+We define the ``Model`` and train it with L-BFGS:
+
+.. code-block:: python
+
+    model = dde.Model(pde_op, net)
+    dde.optimizers.set_LBFGS_options(maxiter=1000)
+    model.compile("L-BFGS")
+    model.train()
+
+Finally, the trained model can be used to predict the solution of the Poisson
+equation. We sample the solution for three random representations of :math:`f`.
+
+.. code-block:: python
+
+    n = 3
+    features = space.random(n)
+    fx = space.eval_batch(features, evaluation_points)
+
+    x = geom.uniform_points(100, boundary=True)
+    y = model.predict((fx, x))
+
+
+![](pideeponet_poisson1d.png)
+
+
+Complete code
+-------------
+
+.. literalinclude:: ../../../examples/operator/pideeponet_1d_poisson.py
+  :language: python

docs/demos/operator/poisson_1d_pideeponet.rst

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+"""Backend supported: tensorflow.compat.v1, tensorflow, pytorch"""
+import deepxde as dde
+import matplotlib.pyplot as plt
+import numpy as np
+
+
+# Poisson equation: -u_xx = f
+def equation(x, y, f):
+    dy_xx = dde.grad.hessian(y, x)
+    return -dy_xx - f
+
+
+# Domain is interval [0, 1]
+geom = dde.geometry.Interval(0, 1)
+
+
+# Zero Dirichlet BC
+def u_boundary(_):
+    return 0
+
+
+def boundary(_, on_boundary):
+    return on_boundary
+
+
+bc = dde.icbc.DirichletBC(geom, u_boundary, boundary)
+
+# Define PDE
+pde = dde.data.PDE(geom, equation, bc, num_domain=100, num_boundary=2)
+
+# Function space for f(x) are polynomials
+degree = 3
+space = dde.data.PowerSeries(N=degree + 1)
+
+# Choose evaluation points
+num_eval_points = 10
+evaluation_points = geom.uniform_points(num_eval_points, boundary=True)
+
+# Define PDE operator
+pde_op = dde.data.PDEOperatorCartesianProd(
+    pde,
+    space,
+    evaluation_points,
+    num_function=100,
+)
+
+# Setup DeepONet
+dim_x = 1
+p = 32
+net = dde.nn.DeepONetCartesianProd(
+    [num_eval_points, 32, p],
+    [dim_x, 32, p],
+    activation="tanh",
+    kernel_initializer="Glorot normal",
+)
+
+# Define and train model
+model = dde.Model(pde_op, net)
+dde.optimizers.set_LBFGS_options(maxiter=1000)
+model.compile("L-BFGS")
+model.train()
+
+# Plot realisations of f(x)
+n = 3
+features = space.random(n)
+fx = space.eval_batch(features, evaluation_points)
+
+x = geom.uniform_points(100, boundary=True)
+y = model.predict((fx, x))
+
+# Setup figure
+fig = plt.figure(figsize=(7, 8))
+plt.subplot(2, 1, 1)
+plt.title("Poisson equation: Source term f(x) and solution u(x)")
+plt.ylabel("f(x)")
+z = np.zeros_like(x)
+plt.plot(x, z, "k-", alpha=0.1)
+
+# Plot source term f(x)
+for i in range(n):
+    plt.plot(evaluation_points, fx[i], "--")
+
+# Plot solution u(x)
+plt.subplot(2, 1, 2)
+plt.ylabel("u(x)")
+plt.plot(x, z, "k-", alpha=0.1)
+for i in range(n):
+    plt.plot(x, y[i], "-")
+plt.xlabel("x")
+
+plt.show()