# On the Forward Invariance of Neural ODEs

arXiv: https://arxiv.org/abs/2210.04763 · HTML: https://mlsanity.com/paper/2210.04763/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 62 references that are in the map.
- First posted on arXiv 10 Oct 2022.

## Builds on (most cited first, 30 of 34)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Control Barrier Function Based Quadratic Programs for Safety Critical Systems](https://mlsanity.com/paper/1609.06408/) (2016)
- [Differentiable MPC for End-to-end Planning and Control](https://mlsanity.com/paper/1810.13400/) (2018)
- [Liquid Time-constant Networks](https://mlsanity.com/paper/2006.04439/) (2020)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [Liquid Structural State-Space Models](https://mlsanity.com/paper/2209.12951/) (2022)
- [Learning Long-Term Dependencies in Irregularly-Sampled Time Series](https://mlsanity.com/paper/2006.04418/) (2020)
- [Stiff Neural Ordinary Differential Equations](https://mlsanity.com/paper/2103.15341/) (2021)
- [Closed-form Continuous-time Neural Models](https://mlsanity.com/paper/2106.13898/) (2021)
- [Neural Certificates for Safe Control Policies](https://mlsanity.com/paper/2006.08465/) (2020)
- [Hypersolvers: Toward Fast Continuous-Depth Models](https://mlsanity.com/paper/2007.09601/) (2020)
- [Control Barrier Functions for Systems with High Relative Degree](https://mlsanity.com/paper/1903.04706/) (2019)
- [Neural Ordinary Differential Equation Control of Dynamics on Graphs](https://mlsanity.com/paper/2006.09773/) (2020)
- [On The Verification of Neural ODEs with Stochastic Guarantees](https://mlsanity.com/paper/2012.08863/) (2020)
- [Causal Navigation by Continuous-time Neural Networks](https://mlsanity.com/paper/2106.08314/) (2021)
- [Safe Optimal Control Using Stochastic Barrier Functions and Deep Forward-Backward SDEs](https://mlsanity.com/paper/2009.01196/) (2020)
- [DyNODE: Neural Ordinary Differential Equations for Dynamics Modeling in Continuous Control](https://mlsanity.com/paper/2009.04278/) (2020)
- [Learning Safe Neural Network Controllers with Barrier Certificates](https://mlsanity.com/paper/2009.09826/) (2020)
- [ShieldNN: A Provably Safe NN Filter for Unsafe NN Controllers](https://mlsanity.com/paper/2006.09564/) (2020)
- [Sparse Flows: Pruning Continuous-depth Models](https://mlsanity.com/paper/2106.12718/) (2021)
- [Forecasting the outcome of spintronic experiments with Neural Ordinary Differential Equations](https://mlsanity.com/paper/2108.02318/) (2021)
- [GoTube: Scalable Stochastic Verification of Continuous-Depth Models](https://mlsanity.com/paper/2107.08467/) (2021)
- [BarrierNet: A Safety-Guaranteed Layer for Neural Networks](https://mlsanity.com/paper/2111.11277/) (2021)

## Built on by (most cited first, 11 of 11)

- [SafeDiffuser: Safe Planning with Diffusion Probabilistic Models](https://mlsanity.com/paper/2306.00148/) (2023)
- [FI-ODE: Certifiably Robust Forward Invariance in Neural ODEs](https://mlsanity.com/paper/2210.16940/) (2022)
- [Safe Neural Control for Non-Affine Control Systems with Differentiable Control Barrier Functions](https://mlsanity.com/paper/2309.04492/) (2023)
- [Certified Robust Invariant Polytope Training in Neural Controlled ODEs](https://mlsanity.com/paper/2408.01273/) (2024)
- [Interpreting Neural Policies with Disentangled Tree Representations](https://mlsanity.com/paper/2210.06650/) (2022)
- [KirchhoffNet: A Scalable Ultra Fast Analog Neural Network](https://mlsanity.com/paper/2310.15872/) (2023)
- [ABNet: Attention BarrierNet for Safe and Scalable Robot Learning](https://mlsanity.com/paper/2406.13025/) (2024)
- [Understanding the Theoretical Foundations of Deep Neural Networks through Differential Equations](https://mlsanity.com/paper/2603.18331/) (2026)
- [TNODEV: Toolbox for Neural ODE Verification](https://mlsanity.com/paper/2606.16567/) (2026)
- [BarrierSteer: LLM Safety via Learning Barrier Steering](https://mlsanity.com/paper/2602.20102/) (2026)
- [SSP: Safety-guaranteed Surgical Policy via Joint Optimization of Behavioral and Spatial Constraints](https://mlsanity.com/paper/2603.07032/) (2026)

## Abstract (arXiv)

We propose a new method to ensure neural ordinary differential equations (ODEs) satisfy output specifications by using invariance set propagation. Our approach uses a class of control barrier functions to transform output specifications into constraints on the parameters and inputs of the learning system. This setup allows us to achieve output specification guarantees simply by changing the constrained parameters/inputs both during training and inference. Moreover, we demonstrate that our invariance set propagation through data-controlled neural ODEs not only maintains generalization performance but also creates an additional degree of robustness by enabling causal manipulation of the system's parameters/inputs. We test our method on a series of representation learning tasks, including modeling physical dynamics and convexity portraits, as well as safe collision avoidance for autonomous vehicles.