A Dynamical Systems view of Feedback Synthesis
arXiv:2608.04172
2026
Dynamics
1 ideas extracted · analyzed Aug 31, 2026
What the math gives to ML
The paper provides a constructive feedback-synthesis mechanism based on lifting a controlled system to a Hamiltonian flow and representing the optimal feedback as an invariant stable manifold. The key transferable asset is the geometric diagnostic that a feedback exists smoothly only where the stable manifold projects diffeomorphically to state space; projection singularities form caustics and predict branch multiplicity, hysteresis, or unavoidable discontinuities. A neural-network implementation can learn a stable-manifold chart and its feedback law while explicitly monitoring the smallest singular value of the state projection. This creates a falsifiable architecture for neural ODE control or optimization: performance and smoothness should degrade sharply as the learned projection approaches a caustic.
Ideas from this paper
✓✓ Beats tuned baseline
2026
Represent a neural controller as the projection of a learned Hamiltonian stable manifold rather than learning a state-to-action map without geometric constraints. Train a manifold chart together with its invariance equation, and reject or branch-switch near points where the manifold projection becomes singular. The resulting controller exposes a measurable boundary between single-valued smooth feedback and multivalued or hysteretic feedback.
Useful8/10
Difficulty6/10
Novelty7/10