# On the Topology of Neural Network Superlevel Sets

arXiv: https://arxiv.org/abs/2603.02973 · HTML: https://mlsanity.com/paper/2603.02973/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 16 references that are in the map.
- First posted on arXiv 3 Mar 2026.

## Builds on (most cited first, 7 of 7)

- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [On Characterizing the Capacity of Neural Networks using Algebraic Topology](https://mlsanity.com/paper/1802.04443/) (2018)
- [Universal Approximation Power of Deep Residual Neural Networks via Nonlinear Control Theory](https://mlsanity.com/paper/2007.06007/) (2020)
- [Topological Expressivity of ReLU Neural Networks](https://mlsanity.com/paper/2310.11130/) (2023)

## Abstract (arXiv)

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.