arXiv:2607.07910 2026 Regularization 1 ideas extracted · analyzed Aug 30, 2026
The paper identifies a complex numerator field whose phase winding is an integer Brouwer degree invariant under positive gauge-like recombinations, equation scaling, and orientation-preserving coordinate changes. This supplies a principled topological invariant for complex-valued neural fields: unlike magnitude losses, it can detect and preserve vortices even when the field is smoothly rescaled or distorted. The most credible ML transfer is a topology-aware auxiliary loss and evaluation metric for networks predicting spatial complex coefficients, vector fields, or PDE solutions, with the discrete charge used to detect spurious vortex creation.
Add a topological loss that preserves the winding number of a complex numerator field predicted by a neural network. The loss is invariant to positive rescaling of the field, so it penalizes vortex creation or destruction rather than harmless amplitude changes.