Universal set of Observables for Forecasting Physical Systems through Causal Embedding

We show how a pair of points can uniquely represent a left-infinite sequence obtained from observations of an underlying dynamical system through a phenomenon called $\mathit{\text{causal}}\mathit{\text{embedding}}$. A driven dynamical system creates such pairs, and a function can be learnt on them that can reconstruct the underlying dynamics as in Takens delay embedding. The approach assures embedding stability unlike Takens delay embedding, and learnability which can be absent while stability can be present in the current reservoir computing framework. This accurately models underlying systems where recent methods like the next-generation reservoir computing fail. We demonstrate results and compare with the other methods including SINDY-PI.