Path signatures of ODE solutions

2025/05/19 by Galuppi, Francesco, Moreno, Giovanni, Santarsiero, Pierpaola
#15A69 #34A34 #60L10 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2505.13234

Abstract

The signature of a path is a sequence of tensors which allows to uniquely reconstruct the path. By employing the geometric theory of nonlinear systems of ordinary differential equations, we find necessary and sufficient algebraic conditions on the signature tensors of a path to be a solution of a given system of ODEs. As an application, we describe in detail the systems of ODEs that describe the trajectories of a vector field, in particular a linear and Hamiltonian one.

Citations

• Rank and symmetries of signature tensors
• An algebraic geometry of paths via the iterated-integrals signature
• SK-Tree: a systematic malware detection algorithm on streaming trees via\n the signature kernel
• Signatures of paths transformed by polynomial maps
• Contact manifolds, Lagrangian Grassmannians and PDEs
• The Signature of a Rough Path: Uniqueness
• Rough paths, Signatures and the modelling of functions on streams
• Natural endomorphisms of shuffle algebras

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