Arbitrary-Order Padé-Closed Anchored Two-Derivative Time Discretizations: $s$ Active Stages, Order $2s$, and $L$-Stability
arXiv:2607.24592
2026
Dynamics
1 ideas extracted · analyzed Aug 31, 2026
What the math gives to ML
The paper constructs implicit two-derivative one-step integrators with s active stages, global order 2s, and an L-stable Padé stability function. The transferable asset is the ability to combine high-order accuracy with strong damping of unresolved fast modes while keeping the number of unknown stage states small. This is relevant to neural ODEs, probability-flow diffusion samplers, and learned dynamical systems whose vector fields can become stiff. The main cost is one total-time-derivative evaluation per stage, which can be obtained with automatic-differentiation JVPs instead of explicitly forming a Jacobian.
Ideas from this paper
△ Mechanism confirmed, baseline not beaten
2026
Replace explicit Runge-Kutta integration in a neural ODE or probability-flow ODE sampler with the anchored two-derivative method. Each stage uses both the neural vector field and its total time derivative, while the coupled implicit solve is designed so the accepted map has an L-stable Padé stability function. The method should allow larger steps on stiff trajectories without amplifying fast decaying modes.
Useful7/10
Difficulty7/10
Novelty7/10