Universal set of Observables for Forecasting Physical Systems through Causal Embedding

Abstract

We demonstrate when and how an entire left-infinite orbit of an underlying dynamical system or observations from such left-infinite orbits can be uniquely represented by a pair of elements in a different space, a phenomenon which we call \textit{causal embedding}. The collection of such pairs is derived from a driven dynamical system and is used to learn a function which together with the driven system would: (i). determine a system that is topologically conjugate to the underlying system (ii). enable forecasting the underlying system's dynamics since the conjugacy is computable and universal, i.e., it does not depend on the underlying system (iii). guarantee an attractor containing the image of the causally embedded object even if there is an error made in learning the function. By accomplishing these we herald a new forecasting scheme that beats the existing reservoir computing schemes that often lead to poor long-term consistency as there is no guarantee of the existence of a learnable function, and overcomes the challenges of stability in Takens delay embedding. We illustrate accurate modeling of underlying systems where previously known techniques have failed.