Learning Aligned Stability in Neural ODEs Reconciling Accuracy with Robustness

Abstract

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.