# Learning Aligned Stability in Neural ODEs Reconciling Accuracy with Robustness

arXiv: https://arxiv.org/abs/2509.21879 · HTML: https://mlsanity.com/paper/2509.21879/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 38 of its 52 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Builds on (most cited first, 30 of 38)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Benchmarking Neural Network Robustness to Common Corruptions and Perturbations](https://mlsanity.com/paper/1903.12261/) (2019)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Adversarial examples in the physical world](https://mlsanity.com/paper/1607.02533/) (2016)
- [Theoretically Principled Trade-off between Robustness and Accuracy](https://mlsanity.com/paper/1901.08573/) (2019)
- [Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks](https://mlsanity.com/paper/2003.01690/) (2020)
- [Square Attack: a query-efficient black-box adversarial attack via random search](https://mlsanity.com/paper/1912.00049/) (2019)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Adversarial Examples: Attacks and Defenses for Deep Learning](https://mlsanity.com/paper/1712.07107/) (2017)
- [Enhancing the Transferability of Adversarial Attacks through Variance Tuning](https://mlsanity.com/paper/2103.15571/) (2021)
- [Torchattacks: A PyTorch Repository for Adversarial Attacks](https://mlsanity.com/paper/2010.01950/) (2020)
- [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)
- [Lipschitz constant estimation of Neural Networks via sparse polynomial optimization](https://mlsanity.com/paper/2004.08688/) (2020)
- [On Robustness of Neural Ordinary Differential Equations](https://mlsanity.com/paper/1910.05513/) (2019)
- [LyaNet: A Lyapunov Framework for Training Neural ODEs](https://mlsanity.com/paper/2202.02526/) (2022)
- [Backdoor Attacks and Countermeasures in Natural Language Processing Models: A Comprehensive Security Review](https://mlsanity.com/paper/2309.06055/) (2023)
- [Stable Neural Stochastic Differential Equations in Analyzing Irregular Time Series Data](https://mlsanity.com/paper/2402.14989/) (2024)
- ["Hey, that's not an ODE": Faster ODE Adjoints via Seminorms](https://mlsanity.com/paper/2009.09457/) (2020)
- [Exploring Misclassifications of Robust Neural Networks to Enhance Adversarial Attacks](https://mlsanity.com/paper/2105.10304/) (2021)
- [Adversarial Robustness in Graph Neural Networks: A Hamiltonian Approach](https://mlsanity.com/paper/2310.06396/) (2023)
- [Adversarial Robustness of Stabilized NeuralODEs Might be from Obfuscated Gradients](https://mlsanity.com/paper/2009.13145/) (2020)
- [IBP Regularization for Verified Adversarial Robustness via Branch-and-Bound](https://mlsanity.com/paper/2206.14772/) (2022)
- [Physics-Informed Neural Network Lyapunov Functions: PDE Characterization, Learning, and Verification](https://mlsanity.com/paper/2312.09131/) (2023)
- [Robust Classification using Contractive Hamiltonian Neural ODEs](https://mlsanity.com/paper/2203.11805/) (2022)
- [Lyapunov-Stable Deep Equilibrium Models](https://mlsanity.com/paper/2304.12707/) (2023)
- [Coupling Graph Neural Networks with Fractional Order Continuous Dynamics: A Robustness Study](https://mlsanity.com/paper/2401.04331/) (2024)
- [Asymmetric Certified Robustness via Feature-Convex Neural Networks](https://mlsanity.com/paper/2302.01961/) (2023)

## Abstract (arXiv)

Despite Neural Ordinary Differential Equations (Neural ODEs) exhibiting intrinsic robustness, existing methods often impose Lyapunov stability for formal guarantees. However, these methods still face a fundamental accuracy-robustness trade-off, which stems from a core limitation: their applied stability conditions are rigid and inappropriate, creating a mismatch between the model's regions of attraction (RoAs) and its decision boundaries. To resolve this, we propose Zubov-Net, a novel framework that unifies dynamics and decision-making. We first employ learnable Lyapunov functions directly as the multi-class classifier, ensuring the prescribed RoAs (PRoAs, defined by the Lyapunov functions) inherently align with a classification objective. Then, for aligning prescribed and true regions of attraction (PRoAs-RoAs), we establish a Zubov-driven stability region matching mechanism by reformulating Zubov's equation into a differentiable consistency loss. Building on this alignment, we introduce a new paradigm for actively controlling the geometry of RoAs by directly optimizing PRoAs to reconcile accuracy and robustness. Theoretically, we prove that minimizing the tripartite loss guarantees consistency alignment of PRoAs-RoAs, non-overlapping PRoAs, trajectory stability, and a certified robustness margin. Moreover, we establish stochastic convex separability with tighter probability bounds and lower dimensionality requirements to justify the convex design in Lyapunov functions.