# Latent Representations of Dynamical Systems: When Two is Better Than One

## TL;DR

This paper demonstrates that using separate latent mappings for present and future states in dynamical systems prediction is theoretically optimal and outperforms traditional single-mapping methods, especially for non-time-reversible systems.

## Contribution

It introduces a novel approach that employs two different latent representations for present and future, challenging the common single-mapping paradigm in dynamical system prediction.

## Key findings

- Two-mapping approach outperforms PCA and single-mapping methods
- Optimality of separate mappings for non-time-reversible systems
- Illustrated with coupled harmonic oscillators with noise and dissipation

## Abstract

A popular approach for predicting the future of dynamical systems involves mapping them into a lower-dimensional "latent space" where prediction is easier. We show that the information-theoretically optimal approach uses different mappings for present and future, in contrast to state-of-the-art machine-learning approaches where both mappings are the same. We illustrate this dichotomy by predicting the time-evolution of coupled harmonic oscillators with dissipation and thermal noise, showing how the optimal 2-mapping method significantly outperforms principal component analysis and all other approaches that use a single latent representation, and discuss the intuitive reason why two representations are better than one. We conjecture that a single latent representation is optimal only for time-reversible processes, not for e.g. text, speech, music or out-of-equilibrium physical systems.