Make AxesArray handle slicing correctly

[Jacob-Stevens-Haas]

Fixes #220

It's long made sense to expose the AxesArray interface and document it. However, when it was created, it wasn't stable enough even for every use case inside pysindy. Indexing can remove, add, and even move around the axes in an array, which meant that properties such as ax_time and n_spatial were unreliable when an array was indexed. FiniteDifference._accumulate() is an example of great code that worked which AxesArrays, but in which AxesArray gave no extra understanding in debugging because the axis labels were incorrect.

This PR makes basic indexing, integer advanced indexing, and boolean array indexing work. It does so by adding a a private attribute that contains both forwards and backwards maps of the axes, and modifying __getitem__ to detect which axes are added, removed, combined, relocated, etc. The PR also expands the ufunc tests dramatically. Item 1 below is probably all that's needed to accept the PR.

• [x] Determine which warnings should be shown depending on axis assignments (check which ones are raised by tests).
• [x] Document this well, including the shortcomings (field access for structured arrays)
• [x] Add issues to make transpose, roll, and reshape work for axesarrays.
• [x] Now that I understand views vs new-from-template, I can probably simplify __array_finalize__. Not sure I need all those conditionals.
• [x] I still cut down the original list of numpy ufunc tests from what numpy itself uses. Now that I understand views vs new-from-template, I can probably go back and make better decisions on what/how we test.

[Jacob-Stevens-Haas]

CI bug related to https://bugs.python.org/issue42233, testing on 3.8/3.9 but it'll be an easy fix either way.

[Jacob-Stevens-Haas]

Heads up, I made AxesArray raise errors for broken axis labels (e.g. labeling a third axis in a 2D array). So numpy functions that change the axes of its inputs raise errors on AxesArray inputs. Previously, they just assigned the same axes as their inputs. I am fixing them one by one. So far I've finished reshape and transpose, did einsum but need to add tests , and need to do linalg.solve and tensordot.

[codecov[bot]]

Codecov Report

Attention: 7 lines in your changes are missing coverage. Please review.

Comparison is base (8341efb) 93.92% compared to head (bc3fbd2) 94.40%. Report is 2 commits behind head on master.

Files	Patch %	Lines
pysindy/utils/axes.py	98.28%	7 Missing :warning:

Additional details and impacted files

```diff @@ Coverage Diff @@ ## master #451 +/- ## ========================================== + Coverage 93.92% 94.40% +0.48% ========================================== Files 37 37 Lines 3602 3989 +387 ========================================== + Hits 3383 3766 +383 - Misses 219 223 +4 ```

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[Jacob-Stevens-Haas]

Issues to add to pysindy repo, discovered during this PR but not for this PR:

• Add einsum tests
• Clear AxesWarnings in tests (about 60 in each test_pysindy and test_sindyc, but far fewer than other warnings) and then replace warning with error
• Add test for calling pysindy with axesarray with different axes - are we smart about comprehend axes?
• Remove default in_ndim and axes in _AxisMapping - args should be required.
• Add support for advanced indexing with labeled AxesArray (e.g. stencil = t[n_stencil]) or provide a better way to do this in developer docs.
• Add support for @ - + * .T etc (currently just given labels from first)
• Replace time variables (e.g. "t") in Differentiation with axis generic variable names
• Clean out unnecesary pieces of documentation API documentation (e.g. pysindy.version module) ... it's a bit out of hand. One page per module or package, and package only with list of links to modules
• Scrub ufunc_override unit
• Decide what should happen with numpy.ndarray.view(AxesArray). Should it fail? Should it accept a second parameter if that's possible?)
• Decide whether we should attempt to make an AxesArray from mixed-type numpy functions (e.g. np.foo((AxesArray, np.ndarray)) or just return ndarray. See__array_function__`. Changing this now breaks pysindy. This may affect the next issue:
• Some nonimplemented functions that create a view, which calls __array_finalize__. Currently that method has ugly conditionals, and it might be cleaner to remove them (but doing so now breaks pysindy).

[Jacob-Stevens-Haas]

Alright @znicolaou, @akaptano, this is ready for your review, if desired! After this I'd suggest we release 2.0.0. One of the main points of this PR was to removes some defaults for AxesArray, so that I'm comfortable fixing issues without breaking backwards compatability.

This PR dramatically changes the AxesArray class to support slicing. This is the most common pysindy operation on arrays. Now that slicing works, AxesArray is stricter about creating arrays/array views. The strictness caused some functions in pysindy to fail; the PR patches FiniteDifference and PDELibrary as well as introduces implementations of common numpy operations on AxesArray (e.g. numpy.linalg.solve).

I'm going to leave this open for at least two weeks. I'd especially appreciate if you could take a look at the changes to the package outside of axes.py, which are many fewer and easier to review. Admittedly, some of these changes only make sense in debugging, when you can see that slices of the spatiotemporal grid carry the correct axis definition.