A metric for comparing complex systems by their dynamics

Comparisons are fundamental to science: experiment against model, one organism against another, a system against itself across time. Because many systems, from brains to climate, are characterized by how they evolve in time, it is a natural goal to compare their dynamics. Dynamical systems comparison is well defined, but has been intractable for nonlinear, high-dimensional, noisy, and partially observed data. As a result, standard comparison methods have focused on the geometry or topology of data. Here we present Dynamical Similarity Analysis (DSA), a class of metrics to compare systems by their temporal evolution. Its foundation is Koopman Operator theory, which recasts nonlinear systems as linear operators. We estimate these operators from data, then compare the operators across systems. The computation is fast, scalable, and robust to noise and partial observation. It is also differentiable. DSA identifies dynamical structure that geometric and topological methods miss. It matches recordings from the head direction circuit to ring attractor models. It shows that macaque motor cortex dynamics for two reaching tasks drift apart across years despite preserved behavior, and that primary motor cortex breaks from premotor cortex as movement begins. As an optimization objective, it induces neural networks to learn never-before hypothesized solutions that run counter to their inductive biases. Thus, DSA transforms the dynamics of a system into an object that can be measured, compared, and optimized.