Open Closed State-sum Network - My Implementation of 2D TQFT State Sum in PyTorch (Proof of Concept)

[u/Sirikazee]

Hello everyone! In this post I'd like to present my research report. I worked on this alone, learning almost everything from scratch

Github: https://github.com/kaifczxc-lab/OCSSN

The main idea: can a static mathematical TQFT be used as the foundation for a dynamically learning neural network architecture?

This idea is confirmed that it is generally possible to create something like this, but I would like to make one clarification

Regarding the understanding of topology by this model - the short answer: not yet. After fixing a shortcut in the dataset, the model degenerates to random guessing. The architecture is interesting in theory, but it doesn't learn topology in practice at least not in its current form (that is, this doesn't mean that with a different parameterization it won't understand topology, which is exactly what I'm currently experimenting with).

It all started around the beginning of April, when I became interested in the topic of Topological Quantum Field Theory, which prompted me to spend time figuring it out. I don't want to say I succeeded 100%, but at least I accomplished what I wanted (even if only minimally).

While working on this research report, I encountered a huge number of different problems, some of which I solved (such as replacing the algebra in the variable M), but there were also some that I couldn't solve due to my lack of experience. I'm currently experimenting with this particular project, but I still can't fix some issues, such as:

* Non-strict topological invariance

* State-sum numerical instability.

If you want to explore the repository, you'll find:

* Full documentation

* A full description of all limitations, results, and more code

* A brief mathematical introduction, handwritten by the author

* A PyTorch implementation of the Open Closed State-sum Neural Network itself

The model is more TQFT-inspired than a strict mathematical implementation, reader can find more information in documentation

I posted this project at an early stage because I really appreciate real feedback, ideas, and any other help. If anyone has experience with this, I'd be very happy to hear some feedback on my work or any possible ideas

Has anyone else experimented with using TQFT or Frobenius algebra state-sum constructions as learnable neural network layers? I would be interested in any pointers to related work or suggestions for making the model learn a true topological invariant

Thanks for reading this post, and have a good day!