# Long-Time Trajectory Approximation via SA-NODEs: Model Predictive and Floquet Strategies

arXiv: https://arxiv.org/abs/2608.10738 · HTML: https://mlsanity.com/paper/2608.10738/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 29 references that are in the map.
- First posted on arXiv 11 Aug 2026.

## Builds on (most cited first, 16 of 16)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Learning Stable Deep Dynamics Models](https://mlsanity.com/paper/2001.06116/) (2020)
- [Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks](https://mlsanity.com/paper/2110.12976/) (2021)
- [Neural ODE control for classification, approximation and transport](https://mlsanity.com/paper/2104.05278/) (2021)
- [LyaNet: A Lyapunov Framework for Training Neural ODEs](https://mlsanity.com/paper/2202.02526/) (2022)
- [Differentiable Multiple Shooting Layers](https://mlsanity.com/paper/2106.03885/) (2021)
- [Multiple shooting for training neural differential equations on time series](https://mlsanity.com/paper/2109.06786/) (2021)
- [Turnpike in optimal control of PDEs, ResNets, and beyond](https://mlsanity.com/paper/2202.04097/) (2022)
- [Learning Complex Motion Plans using Neural ODEs with Safety and Stability Guarantees](https://mlsanity.com/paper/2308.00186/) (2023)
- [Sparsity in long-time control of neural ODEs](https://mlsanity.com/paper/2102.13566/) (2021)
- [Stability and Convergence of a Randomized Model Predictive Control Strategy](https://mlsanity.com/paper/2211.05463/) (2022)
- [Deep Neural ODE Operator Networks for PDEs](https://mlsanity.com/paper/2510.15651/) (2025)
- [Representation and Regression Problems in Neural Networks: Relaxation, Generalization, and Numerics](https://mlsanity.com/paper/2412.01619/) (2024)
- [Universal Approximation Theorems for Dynamical Systems with Infinite-Time Horizon Guarantees](https://mlsanity.com/paper/2602.08640/) (2026)
- [Designing two-dimensional limit-cycle oscillators with prescribed trajectories and phase-response characteristics](https://mlsanity.com/paper/2301.07237/) (2023)
- [Activation Saturation and Floquet Spectrum Collapse in Neural ODEs](https://mlsanity.com/paper/2604.00543/) (2026)

## Abstract (arXiv)

We study the approximation of dynamical systems by semi-autonomous neural ordinary differential equations (SA-NODEs) over long time horizons. For a single network trained on the whole horizon, the available error bound deteriorates double exponentially in the horizon length. We develop two training strategies that avoid this barrier, each built on a reset of the state. The model predictive strategy partitions the horizon adaptively and restarts every window from observed data: when training meets a prescribed tolerance on every window, the composite model meets it uniformly in time, with a parameter budget linear in the horizon for targets with a bounded, uniformly regular reachable tube. The Floquet strategy addresses autonomous targets with a stable limit cycle and uses no data at deployment: a certified contraction of the learned return map confines the error to linear growth in the number of elapsed periods. For the time-periodic architecture we deploy, the scalar certificate degenerates; we prove instead a uniform-in-time orbital guarantee whose hypotheses are measured on the trained model, and an obstruction showing that, for an exactly periodic learned field, small one-period error and a contracting stroboscopic map cannot hold at once. Numerical experiments on four benchmarks confirm the predicted error laws and measure the hypotheses of every guarantee.