# Capabilities and Limitations of Time-lagged Autoencoders for Slow Mode Discovery in Dynamical Systems

## TL;DR

This paper analyzes the capabilities and limitations of time-lagged autoencoders in discovering slow modes in dynamical systems, providing theoretical bounds and numerical comparisons with SRVs.

## Contribution

It offers the first rigorous theoretical analysis of nonlinear TAEs and compares their performance with SRVs in identifying slow modes.

## Key findings

- TAEs learn a mixture of slow and maximum variance modes
- TAEs can fail to identify the slowest mode in certain systems
- SRVs outperform TAEs in some cases of slow mode discovery

## Abstract

Time-lagged autoencoders (TAEs) have been proposed as a deep learning regression-based approach to the discovery of slow modes in dynamical systems. However, a rigorous analysis of nonlinear TAEs remains lacking. In this work, we discuss the capabilities and limitations of TAEs through both theoretical and numerical analyses. Theoretically, we derive bounds for nonlinear TAE performance in slow mode discovery and show that in general TAEs learn a mixture of slow and maximum variance modes. Numerically, we illustrate cases where TAEs can and cannot correctly identify the leading slowest mode in two example systems: a 2D "Washington beltway" potential and the alanine dipeptide molecule in explicit water. We also compare the TAE results with those obtained using state-free reversible VAMPnets (SRVs) as a variational-based neural network approach for slow modes discovery, and show that SRVs can correctly discover slow modes where TAEs fail.