Inverse spectral perspective on dynamics over networks constraints, symmetries and emergent models

Описание: The Koopman operator framework provides a linear description of nonlinear dynamical systems through the spectral properties of operators acting on infinite-dimensional spaces of observables. Despite substantial computational advances and the development of data-driven approaches, extracting Koopman eigenvalues and eigenfunctions analytically remains notoriously difficult, except for a limited class of low-dimensional or linear systems. This difficulty is further exacerbated for nonlinear dynamics defined over large networks, where interactions are encoded by weighted, directed, and possibly signed graphs, leading to high-dimensional state spaces and a number of dynamical parameters that typically scales quadratically with the number of nodes.
In this seminar, I adopt a complementary inverse spectral perspective. Rather than computing Koopman spectra from a given dynamics, I consider the reverse problem: imposing Koopman eigenfunctions (and possibly eigenvalues) a priori, and asking which classes of vector fields and network structures are compatible with this spectral data. For structured families of dynamics—such as Kuramoto and Sakaguchi-type phase oscillator networks—this approach yields explicit analytical constraints on both the dynamical parameters and the underlying graphs that allow the Koopman generator to admit the prescribed eigenfunctions. These constraints uncover hidden symmetries, conservation laws, and admissible network motifs that are difficult to identify through forward spectral analysis. I then formulate a broader inverse problem: given partial Koopman spectral information, characterize all networked vector fields that realize it. This viewpoint naturally gives rise to several families of networked dynamical systems, including continuous-time recurrent neural networks and Lotka–Volterra models on graphs. This seminar is based on joint work with Antoine Allard, Benjamin Claveau, and Vincent Thibeault.