Lift the Conley index from latent dynamics to the original system

Establish verifiable conditions under which Conley index information computed for a learned latent dynamics map can be lifted to the corresponding original high-dimensional dynamical system.

Background

The paper proves that, under a local approximate semiconjugacy condition, attracting blocks for the latent map can be lifted to attracting blocks for the original dynamics, thereby guaranteeing the existence of corresponding attractors. The examples, however, frequently recover matching Conley index information even when the sufficient residual condition is violated.

The authors explicitly identify the transfer of Conley index information—not merely the existence of attractors—as unresolved. A solution would provide a rigorous explanation for when topological invariants computed from a low-dimensional learned representation remain valid for the original high-dimensional system.