Some properties of high-order nonstandard multistep multistage methods

What the math gives to ML

The paper develops nonstandard multistep-multistage and general linear ODE methods that replace the raw step size by a denominator function while retaining the formal order of standard schemes and preserving qualitative properties such as boundedness for arbitrary positive step sizes. The transferable asset is the combination of a block-state recurrence, a positive bounded effective step, and invariant-set assumptions that prevent numerical trajectories from leaving a safe region. A practical neural-network use is to build implicit or semi-implicit residual blocks whose internal dynamics use this nonstandard recurrence, then test whether large-depth or large-step networks remain stable without reducing the nominal step size.