Beyond DSA: Conjugacy-based Comparison of Dynamical Systems

What the math gives to ML

The paper identifies a concrete failure mode of orthogonal Koopman alignment: coordinate changes that preserve the dynamics can be strongly non-orthogonal, while orthogonal operator similarity can also occur without a valid state-space correspondence. The transferable asset is the conjugacy equation h\circ f=g\circ h, which supplies a trajectory-level consistency constraint rather than an arbitrary latent-space alignment objective. A practical neural implementation is to use an explicitly invertible learned state map and train it jointly with latent dynamics so that mapped one-step transitions commute with the two dynamics. This can regularize latent dynamical models, compare independently trained sequence models, or transfer representations without forcing Euclidean or orthogonal coordinates to match.