Improve demo file.

docs/demos/operator/pideeponet_poisson1d.rst

@@ -14,8 +14,7 @@ for the one-dimensional Poisson problem
 
 with zero Dirichlet boundary conditions :math:`u(0) = u(1) = 0`.
 
-The source term :math:`f` is supposed to be an arbitrary continuous function,
-we choose polynomials of degree three.
+The source term :math:`f` is supposed to be an arbitrary continuous function.
 
 
 Implementation
@@ -51,6 +50,8 @@ First, we define the PDE with boundary conditions and the domain:
 
 
 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.
@@ -72,8 +73,8 @@ the loss function.
 
 
 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 the trunk net is a fully connected neural network of size ``[dim_x, 32, p]``.
+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
 
@@ -87,7 +88,7 @@ and the the trunk net is a fully connected neural network of size ``[dim_x, 32,
     )
 
 
-We define a ``Model`` and train it with L-BFGS (pytorch) or Adam:
+We define the ``Model`` and train it with either L-BFGS (pytorch) or Adam:
 
 .. code-block:: python
 
@@ -101,7 +102,17 @@ We define a ``Model`` and train it with L-BFGS (pytorch) or Adam:
         model.train(iterations=10000)
 
 Finally, the trained model can be used to predict the solution of the Poisson
-equation for any admissible source term :math:`f`.
+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)