Universal Set of Observables for Forecasting Physical Systems Through Causal Embedding.

Area of Science:

• Dynamical Systems Theory
• Time Series Analysis
• Machine Learning

Background:

• Takens delay embedding is a common method for reconstructing dynamical systems from time-series data.
• Existing methods like next-generation reservoir computing can lack stability or learnability.

Purpose of the Study:

• To introduce a novel method, causal embedding, for representing left-infinite sequences from dynamical systems.
• To demonstrate the stability and learnability advantages of causal embedding over existing techniques.

Main Methods:

• Utilizing a pair of points to uniquely represent a left-infinite sequence.
• Learning a function on these point pairs to reconstruct underlying dynamics.
• Comparing causal embedding with Takens delay embedding, reservoir computing, and SINDY-PI.

Main Results:

• Causal embedding provides a unique representation of sequences derived from driven dynamical systems.
• The method ensures embedding stability, a limitation of Takens delay embedding.
• It achieves learnability, which can be absent in other frameworks, and outperforms next-generation reservoir computing.

Conclusions:

• Causal embedding offers a robust and accurate approach for modeling dynamical systems.
• This method addresses key limitations in current time-series analysis and dynamical system reconstruction techniques.