Add PI-DeepONet example for 1D Poisson equation.

[samuelburbulla]

This PR proposes an example for training a physics-informed DeepONet with DeepXDE.
The example solves a simple one-dimensional Poisson equation where the source term is the input to the solution operator.

Related to #1308.

[samuelburbulla]

I improved this PR a little more:

• Incorporate some fixes for using PyTorch backend
• Add auxiliary_variables in PyTorch.NN
• Use L-BFGS for training the DeepONet

Here's a plot of the result:

[lululxvi]

I suggest splitting this PR into two separate PRs

1. the demo example code;
2. Pytorch backed code.

[samuelburbulla]

Thanks, @lululxvi, for your review!

This PR is now split into:

• Add PI-DeepONet example for 1D Poisson equation. #1311 (this one) including the example only
• Add auxiliary_variables for pytorch. #1318 including the PyTorch backend modifications

I have found better solutions for the points mentioned, see comments above.

[sibyjackgrove]

@samuelburbulla Thanks for making this very useful example. @lululxvi would you be merging this soon?

[lululxvi]

Which backends does the new code support? The existing code supports tensorflow 1.x.

[samuelburbulla]

Which backends does the new code support? The existing code supports tensorflow 1.x.

This example is tested with tensorflow.compat.v1, tensorflow, and pytorch.

[lululxvi]

I have added a few PI-DeepONet examples, see https://deepxde.readthedocs.io/en/latest/demos/operator.html#pi-deeponet

Could you split this PR into two PRs:

1. only modify the source code;
2. add new example

[samuelburbulla]

I have added a few PI-DeepONet examples

Nice!

Could you split this PR into two Pls

Done, see #1379.

[samuelburbulla]

Rebased on master.

Together with #1403, this example works with all mentioned backends.

Is there anything else that needs to be done here?

[lululxvi]

Reformat via black https://pypi.org/project/black/