Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generic controllability of equivariant systems and applications to particle systems and neural networks

Andrei Agrachev, Cyril Letrouit

TL;DR

It is proved that generic systems with symmetries are controllable in this sense, which has several applications, for instance, generic controllability of particle systems when the kernel of interaction between particles plays the role of a mean-field control.

Abstract

There exist many examples of systems which have some symmetries, and which one may monitor with symmetry preserving controls. Since symmetries are preserved along the evolution, full controllability is not possible, and controllability has to be considered inside sets of states with same symmetries. We prove that generic systems with symmetries are controllable in this sense. This result has several applications, for instance: (i) generic controllability of particle systems when the kernel of interaction between particles plays the role of a mean-field control; (ii) generic controllability for families of vector fields on manifolds with boundary; (iii) universal interpolation for neural networks architectures with "generic" self attention-type layers - a type of layers ubiquitous in recent neural networks architectures, e.g., in the Transformers architecture. The tools we develop could help address various other questions of control of equivariant systems.

Generic controllability of equivariant systems and applications to particle systems and neural networks

TL;DR

It is proved that generic systems with symmetries are controllable in this sense, which has several applications, for instance, generic controllability of particle systems when the kernel of interaction between particles plays the role of a mean-field control.

Abstract

There exist many examples of systems which have some symmetries, and which one may monitor with symmetry preserving controls. Since symmetries are preserved along the evolution, full controllability is not possible, and controllability has to be considered inside sets of states with same symmetries. We prove that generic systems with symmetries are controllable in this sense. This result has several applications, for instance: (i) generic controllability of particle systems when the kernel of interaction between particles plays the role of a mean-field control; (ii) generic controllability for families of vector fields on manifolds with boundary; (iii) universal interpolation for neural networks architectures with "generic" self attention-type layers - a type of layers ubiquitous in recent neural networks architectures, e.g., in the Transformers architecture. The tools we develop could help address various other questions of control of equivariant systems.
Paper Structure (22 sections, 16 theorems, 78 equations)

This paper contains 22 sections, 16 theorems, 78 equations.

Table of Contents

  1. Introduction and main results
  2. A motivating example
  3. Main results
  4. Applications
  5. Open questions
  6. Bibliography
  7. Organization of the paper.
  8. Acknowledgments.
  9. Applications
  10. Manifolds with boundary
  11. Spectrum of matrices
  12. Particle systems
  13. Universal interpolation with generic self-attention layers
  14. Control of quantum systems with symmetries
  15. Proof of Theorem \ref{['t:genequiv']}
  16. ...and 7 more sections

Key Result

Theorem 1.3

There exists for any $k\geq 2$ a set of $k$-uples $(X_1,\ldots,X_k)\in ({\rm Vec}^G(M_G))^k$ which is residual in $({\rm Vec}^G(M_G))^k$ and for which controllability holds in strata.

Theorems & Definitions (31)

  • Example 1.1
  • Definition 1.2: Controllability in strata
  • Theorem 1.3
  • Definition 1.4: Simultaneous controllability in strata
  • Theorem 1.5
  • Corollary 2.1: Generic control on manifolds with boundary
  • Corollary 2.2: Generic control of the spectrum of symmetric matrices
  • Corollary 2.3: Generic control of particle systems
  • Corollary 2.4
  • Example 3.1
  • ...and 21 more