Data-driven modelling of nonlinear dynamics by barycentric coordinates and memory
This work addresses the challenge of data-driven modeling for nonlinear dynamics, which is incremental as it builds on existing projection and memory techniques.
The authors tackled the problem of modeling nonlinear dynamical systems from data by introducing a method that projects states onto convex polytopes and uses memory to retain information, resulting in stable models capable of reproducing chaotic dynamics and complex attractors.
We present a numerical method to model dynamical systems from data. We use the recently introduced method Scalable Probabilistic Approximation (SPA) to project points from a Euclidean space to convex polytopes and represent these projected states of a system in new, lower-dimensional coordinates denoting their position in the polytope. We then introduce a specific nonlinear transformation to construct a model of the dynamics in the polytope and to transform back into the original state space. To overcome the potential loss of information from the projection to a lower-dimensional polytope, we use memory in the sense of the delay-embedding theorem of Takens. By construction, our method produces stable models. We illustrate the capacity of the method to reproduce even chaotic dynamics and attractors with multiple connected components on various examples.