Limitations and future work
What the paper states as theoretical limits of the current design, and the two directions of ongoing work.
What the method cannot promise yet, and what the authors plan next.
Honest limits
This is a first version, tested on four problems. Two possible weak spots are named, both still theoretical. With very large datasets the formula might need so many folds that the arithmetic overflows. And when a problem's rule has a great many exceptions, connecting the dots with straight segments might not be flexible enough, leaving training stuck in a poor solution. Whether and when these actually happen is part of the ongoing work.
What comes next
Two directions. On the maths side, try alternative formulas for the map and the vortices; some may generalise even better. On the engineering side, a fast library written in C that uses the parallel instructions of modern processors, roughly eight times faster, followed by a graphics-card version, so that the method can be tried on medium and large datasets.
Limitations
The main limitation nowadays may occur, in theory, with big datasets. In those cases the parameters \(S\) and \(R\) of the leietanic function (14) could become large and its computation could generate NaN values. Another theoretical limitation may occur for big datasets with many exceptions to the main intrinsic logic: the leietanic function would require a large number of local minima and maxima, and linear interpolation may not be sufficient to replicate them properly. These limitations are theoretical so far; part of the future work is to identify whether they really happen, and under what circumstances.
Future work
Better generalisation
There exist alternative mathematical formulations for the leietanic function and for the vortex scalar fields. A reasonable hypothesis is that some alternative designs or implementations offer better generalisation and hence higher prediction accuracy.
Larger datasets
Improvements to the model would naturally scale the size of the problems it can handle, but computation times with medium and large datasets must be optimised. The plan is a speed-optimised Python library with a core in a high-performance language such as C, making effective use of the CPU cache (L1, L2 and L3) to reduce memory access times and of the AVX2 and AVX-512 vector extensions to process eight 32-bit floats in parallel (SIMD). Together these are expected to speed computation by roughly eight times. The library will then be ported to GPUs: Adam permits many calculations to run concurrently, and the leietanic function includes several linear-algebra operations that parallelise well, increasingly so for larger \(F\), \(S\), \(R\) or \(D\).