From finite vector field data to combinatorial dynamical systems in the sense of Forman

We introduce a three-step method to construct on a simplicial complex a combinatorial dynamical system in the sense of Forman from vector field data by means of a linear minimization problem, where the solution to this problem induces an admissible matching for the dynamical system. We show that the matrix of the minimization problem is unimodular, allowing us to relax the problem from binary-valued to real-valued and permitting its resolution in polynomial time. We demonstrate the effectiveness of the method on the Lotka–Volterra model and on the Lorenz attractor model. We also describe three potential extensions to our method: how barycentric subdivision can be applied to the simplicial complex to increase the resolution and obtain a solution that better fits the underlying dynamics of the data, how to add constraints to the minimization problem to fix the number of critical simplices, and how to add constraints to obtain a solution that induces a gradient matching.