# PAC bounds of continuous Linear Parameter-Varying systems related to neural ODEs

arXiv: https://arxiv.org/abs/2307.03630 · HTML: https://mlsanity.com/paper/2307.03630/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 43 references that are in the map.
- First posted on arXiv 7 Jul 2023.

## Builds on (most cited first, 21 of 21)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Stable Neural ODE with Lyapunov-Stable Equilibrium Points for Defending Against Adversarial Attacks](https://mlsanity.com/paper/2110.12976/) (2021)
- [Finite Sample Analysis of Stochastic System Identification](https://mlsanity.com/paper/1903.09122/) (2019)
- [Learning Linear Dynamical Systems with Semi-Parametric Least Squares](https://mlsanity.com/paper/1902.00768/) (2019)
- [Logarithmic Regret Bound in Partially Observable Linear Dynamical Systems](https://mlsanity.com/paper/2003.11227/) (2020)
- [Stability Analysis using Quadratic Constraints for Systems with Neural Network Controllers](https://mlsanity.com/paper/2006.07579/) (2020)
- [LyaNet: A Lyapunov Framework for Training Neural ODEs](https://mlsanity.com/paper/2202.02526/) (2022)
- [Logarithmic Regret for Adversarial Online Control](https://mlsanity.com/paper/2003.00189/) (2020)
- [Do Residual Neural Networks discretize Neural Ordinary Differential Equations?](https://mlsanity.com/paper/2205.14612/) (2022)
- [Framing RNN as a kernel method: A neural ODE approach](https://mlsanity.com/paper/2106.01202/) (2021)
- [Adversarial Robustness of Stabilized NeuralODEs Might be from Obfuscated Gradients](https://mlsanity.com/paper/2009.13145/) (2020)
- [Generalization bounds for neural ordinary differential equations and deep residual networks](https://mlsanity.com/paper/2305.06648/) (2023)
- [Robust Classification using Contractive Hamiltonian Neural ODEs](https://mlsanity.com/paper/2203.11805/) (2022)
- [Learning Recurrent Neural Net Models of Nonlinear Systems](https://mlsanity.com/paper/2011.09573/) (2020)
- [Reduced-order modeling of LPV systems in the Loewner framework](https://mlsanity.com/paper/2104.10767/) (2021)
- [LPV Modeling of Nonlinear Systems: A Multi-Path Feedback Linearization Approach](https://mlsanity.com/paper/2103.14622/) (2021)
- [Learning Stable and Robust Linear Parameter-Varying State-Space Models](https://mlsanity.com/paper/2304.01828/) (2023)

## Abstract (arXiv)

We consider the problem of learning Neural Ordinary Differential Equations (neural ODEs) within the context of Linear Parameter-Varying (LPV) systems in continuous-time. LPV systems contain bilinear systems which are known to be universal approximators for non-linear systems. Moreover, a large class of neural ODEs can be embedded into LPV systems. As our main contribution we provide Probably Approximately Correct (PAC) bounds under stability for LPV systems related to neural ODEs. The resulting bounds have the advantage that they do not depend on the integration interval.