Neural Networks and ODEs Compute Primitive Recursion via Dynamics, Not Composition
Bournez proves recurrent ReLU networks, polynomial ODEs, and discrete maps all express primitive recursive functions through continuous-time trajectories rather than symbolic subroutine chaining.
Recurrent ReLU networks, polynomial ODEs, and polynomial maps equivalently compute primitive recursion via bounded iteration and continuous dynamics.
- — All three frameworks—RNNs, polynomial ODEs, discrete maps—express primitive recursive functions through fixed dynamical systems.
- — Composition emerges from trajectory evolution, not from explicit closure rules or subroutine calls.
- — Time bounds are themselves primitive recursive; inputs are raw integer vectors.
- — Polynomial ODEs robustly perform rounding and phase selection via continuous flow; fixed polynomial maps cannot.
- — ReLU gates enable exact branching; step-size parameters in discrete maps recover continuous-time benefits with discretization trade-offs.
- — These models shape dynamical trajectories through clocks and error correction, structurally unlike symbolic programming.
- — Framework enables studying subrecursive hierarchies by restricting time, polynomial degree, or discretization resources.
Astrobobo tool mapping
- Reading Queue Add this paper to a queue tagged 'theory' and schedule 45 min to work through the main theorem and one equivalence proof (RNN or ODE).
- Knowledge Capture After reading, record the three key asymmetries (polynomial maps lack rounding, ReLU lacks continuous control, ODEs lack step-size discretion) as a comparison table for future reference.
- Focus Brief Summarize in one page: 'How does my current neural architecture exploit or ignore dynamical properties?' Use the paper's framework to audit your model.
Frequently asked
- Yes, according to Bournez's theorem. Any primitive recursive function can be compiled into a fixed polynomial ODE system with bounded iteration. The ODE operates on real-valued states and uses continuous-time flow to perform rounding and phase selection robustly, though the time bound itself must be primitive recursive.
cite ▸
APA
Olivier Bournez. (2026, April 28). Neural Networks and ODEs Compute Primitive Recursion via Dynamics, Not Composition. Astrobobo Content Engine (rewrite of arxiv/cs.LG). https://astrobobo-content-engine.vercel.app/article/neural-networks-and-odes-compute-primitive-recursion-via-dynamics-not-compositio-104fc8
MLA
Olivier Bournez. "Neural Networks and ODEs Compute Primitive Recursion via Dynamics, Not Composition." Astrobobo Content Engine, 28 Apr 2026, https://astrobobo-content-engine.vercel.app/article/neural-networks-and-odes-compute-primitive-recursion-via-dynamics-not-compositio-104fc8. Based on "arxiv/cs.LG", https://arxiv.org/abs/2604.24356.
BibTeX
@misc{astrobobo_neural-networks-and-odes-compute-primitive-recursion-via-dynamics-not-compositio-104fc8_2026,
author = {Olivier Bournez},
title = {Neural Networks and ODEs Compute Primitive Recursion via Dynamics, Not Composition},
year = {2026},
url = {https://astrobobo-content-engine.vercel.app/article/neural-networks-and-odes-compute-primitive-recursion-via-dynamics-not-compositio-104fc8},
note = {Astrobobo rewrite of arxiv/cs.LG, https://arxiv.org/abs/2604.24356},
}