Remainders of generalised Taylor expansions and a priori bounds for rough differential equations

What the math gives to ML

The paper develops a constructive, tree-indexed generalized Taylor calculus for rough differential equations, with remainder formulas expressed through products of gradients of elementary differentials. Its transferable asset is the organization of noncommutative vector-field compositions into planar binary trees together with cancellation-aware remainder control under Lipschitz assumptions on elementary differentials. A promising neural application is a rough-signature or neural-CDE block that replaces Euler updates by an adaptively truncated tree expansion, improving accuracy and stability on irregularly sampled or highly oscillatory streams at a controlled number of Jacobian-vector products.