# Improving the robustness of neural ODEs with minimal weight perturbation

arXiv: https://arxiv.org/abs/2501.10740 · HTML: https://mlsanity.com/paper/2501.10740/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jan 2025.

## Built on by (most cited first, 4 of 4)

- [Approximation properties of neural ODEs](https://mlsanity.com/paper/2503.15696/) (2025)
- [Manifold-based Algorithms for the Hadamard Decomposition](https://mlsanity.com/paper/2605.28980/) (2026)
- [On the Role of Strain and Vorticity in Numerical Integration Error for Flow Matching](https://mlsanity.com/paper/2605.06680/) (2026)
- [Learning Aligned Stability in Neural ODEs Reconciling Accuracy with Robustness](https://mlsanity.com/paper/2509.21879/) (2025)

## Abstract (arXiv)

We propose a method to enhance the stability of a neural ordinary differential equation (neural ODE) by reducing the maximum error growth subsequent to a perturbation of the initial value. Since the stability depends on the logarithmic norm of the Jacobian matrix associated with the neural ODE, we control the logarithmic norm by perturbing the weight matrices of the neural ODE by a smallest possible perturbation (in Frobenius norm). We do so by engaging an eigenvalue optimisation problem, for which we propose a nested two-level algorithm. For a given perturbation size of the weight matrix, the inner level computes optimal perturbations of that size, while - at the outer level - we tune the perturbation amplitude until we reach the desired uniform stability bound. We embed the proposed algorithm in the training of the neural ODE to improve its robustness to perturbations of the initial value, as adversarial attacks. Numerical experiments on classical image datasets show that an image classifier including a neural ODE in its architecture trained according to our strategy is more stable than the same classifier trained in the classical way, and therefore, it is more robust and less vulnerable to adversarial attacks.