# Generic and Isometric Embeddings in Reservoir Computers

## TL;DR

This paper proves that reservoir computers can embed input system attractors in a way that preserves their structure, with explicit constructions for linear systems and conditions for high-dimensional reservoirs.

## Contribution

It establishes conditions for generalized and isometric embeddings in reservoir systems, linking topological and metric preservation to reservoir dimension and linearity.

## Key findings

- Reservoir systems can admit topological embeddings of input attractors.
- High-dimensional reservoirs enable isometric embeddings.
- Explicit isometric embeddings are constructed for linear systems.

## Abstract

We prove that a generic reservoir system admits a generalized synchronization that is a topological embedding of the input system's attractor. We also prove that for sufficiently high reservoir dimension (given by Nash's embedding theorem) there exists an isometric embedding generalized synchronization. The isometric embedding can be constructed explicitly when the reservoir system and source dynamics are linear.