The Hodge structure of Berry-phase transport: topology, geometry, and noise

What the math gives to ML

The paper supplies an explicit orthogonal decomposition of differential forms on a torus into harmonic, exact, and co-exact components, with each component carrying different global or local structure. This is transferable to neural ODEs, flow-matching models, and periodic/vector-field networks by constraining a learned field to separate constant topological transport from curl-free and divergence-free dynamics. The key engineering asset is that the decomposition is computable by FFT projections, giving an interpretable architectural factorization rather than an additional expensive learned module. A useful first test is whether sector-specific parameterization improves trajectory stability or sample quality at equal parameter count, especially when target dynamics contain both conservative and rotational components.