Finite-Rank Lie Algebroids for Singular Foliations of Prescribed Vanishing Order

What the math gives to ML

The paper gives an exact finite generating system for smooth vector fields whose coefficients vanish to a prescribed order at a designated point. This can be transferred into neural dynamical models by factoring the vector field through degree-k monomials, enforcing an equilibrium with zero derivatives through order k-1 without adding a penalty term. The construction is closed under Lie brackets, which supports repeated composition of constrained vector-field blocks. The most direct test is a Neural ODE or residual dynamics model with a known rest state, measuring local drift, rollout stability, and parameter overhead against an unconstrained MLP.