A class of mesh-free algorithms for mathematical finance, machine learning and fluid dynamics

Abstract

We introduce a numerical methodology that applies to a broad class of partial differential equations and discrete models, and is referred to here as the transport-based mesh-free method. It led us to the development of several numerical algorithms which are now implemented in a Python library, called CodPy. We propose 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 and material science. Our strategy is relevant when a large number of dimensions or degrees of freedom are present, as is the case in mathematical finance, fluid dynamics, and machine learning. We cover here primarily the Fokker-Planck- Kolmogorov system (relevant for mathematical finance and fluid dynamics) and a class of 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.