Tensorflow 1.x backend: multiple outputs extension of DeepONet

[vl-dud]

This pull request suggests an enhancement to DeepONet, addressing its current limitation of producing only a scalar output, which restricts its applicability in various tasks. The functionality of handling multiple outputs has been added to both the DeepONet and DeepONetCartesianProd classes. This extension is currently implemented solely for the TensorFlow 1.x backend. Also the PR includes an example of usage for a Poisson-like 2D problem.

[lululxvi]

There are different ways for multiple outputs. Can you describe how you implement it?

[vl-dud]

I chose the method presented in the article https://arxiv.org/abs/2106.05384

The formula (1)

is replaced by (2)

for i = 1, . . . , n. n is the number of outputs.

Thereby the original implementation of formula (1) has been extracted into a method _build_vanilla_deeponet. _build_vanilla_deeponet is now invoked the necessary number of times, aligning with the number of outputs:

if self.output_count == 1:
    self.y = self._build_vanilla_deeponet()
else:
    ys = []
    for _ in range(0, self.output_count):
        ys.append(self._build_vanilla_deeponet())
    self.y = tf.stack(ys, axis=2)

[lululxvi]

In https://doi.org/10.1016/j.cma.2022.114778, we introduced a few ways for multiple outputs. But it is unclear yet which one is better.

[vl-dud]

My implementation closely resembles the first option outlined in section 3.1.6:

1. The simplest approach is that we can directly use n independent DeepONets, and each DeepONet outputs only one function.

It may be worth considering a future enhancement where all methods are implemented, allowing users to choose approach using a parameter in init().

[vl-dud]

@lululxvi I have added the approaches described in https://doi.org/10.1016/j.cma.2022.114778

[vl-dud]

@mitchelldaneker I merged your branch but there are some issues identified by Codacy. Please check https://app.codacy.com/gh/lululxvi/deepxde/pullRequest?prid=12314151

[mitchelldaneker]

@mitchelldaneker I merged your branch but there are some issues identified by Codacy. Please check https://app.codacy.com/gh/lululxvi/deepxde/pullRequest?prid=12314151

I have made changes addressing these issues via a new pull request. Please take a look. I moved the input transform and output transform into the DeepONet call functions and out of the DeepONetStrategy call functions. I have also fixed an unresolved issue I missed in the triple.py file for multiple losses.

[vl-dud]

@lululxvi the code is now ready for your review

[lululxvi]

This PR becomes very long now and hard to review. Is it possible to split this PR into a few smaller ones?

[vl-dud]

Divided into #1532 and #1533