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

arXiv: https://arxiv.org/abs/2404.08289 · HTML: https://mlsanity.com/paper/2404.08289/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Apr 2024.

## Built on by (most cited first, 9 of 9)

- [Transformers are Universal In-context Learners](https://mlsanity.com/paper/2408.01367/) (2024)
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- [The Intrinsic Dimension of Prompts in Internal Representations of Large Language Models](https://mlsanity.com/paper/2501.10573/) (2025)
- [Transformers through the lens of support-preserving maps between measures](https://mlsanity.com/paper/2509.25611/) (2025)
- [Exact Sequence Interpolation with Transformers](https://mlsanity.com/paper/2502.02270/) (2025)
- [A unified framework for establishing the universal approximation of transformer-type architectures](https://mlsanity.com/paper/2506.23551/) (2025)
- [Universal interpolation for deep residual self-attention networks](https://mlsanity.com/paper/2610.01981/) (2026)
- [Deep learning from a control vantage point](https://mlsanity.com/paper/2610.14385/) (2026)

## Abstract (arXiv)

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.