A Class of Mesh-free Algorithms for Finance, Machine Learning and Fluid Dynamics

Résumé

We introduce a numerical methodology which applies to a broad class of partial differential equations and discrete models, and is referred to here as the {\sl transport-based mesh-free method}. It led us to several numerical algorithms which are now implemented in a Python library, called CodPy. We develop a mesh-free discretization technique based on the (so-called RKHS) theory of reproducing kernels and the theory of transport mappings, in a way that is reminiscent of Lagrangian methods in computational fluid dynamics. The strategy is relevant when a large number of dimensions or degrees of freedom are present, as is the case in mathematical finance and machine learning, but is also applicable in fluid dynamics. We present our algorithms primarily for the Fokker-Planck-Kolmogorov system of mathematical finance and for neural networks based on support vector machines. The proposed algorithms are nonlinear in nature and enjoy quantitative error estimates based on the notion of discrepancy error, which allow one to evaluate the relevance and accuracy of given data and numerical solutions. Joint with Jean Marc Mercier