On the Topology of Neural Network Superlevel Sets
Abstract
We show that neural networks with activations satisfying a Riccati-type ordinary differential equation condition, an assumption arising in recent universal approximation results in the uniform topology, produce Pfaffian outputs on analytic domains with format controlled only by the architecture. Consequently, superlevel sets, as well as Lie bracket rank drop loci for neural network parameterized vector fields, admit architecture-only bounds on topological complexity, in particular on total Betti numbers, uniformly over all weights.
Fields of Study & Subjects
Topological and Geometric Data Analysis
Neural Networks and Applications
Advanced Differential Equations and Dynamical Systems
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 3 March 2026
- Language
- EN
- Document type
- Preprint