We introduce a numerical methodology, referred to as the transport-based mesh-free method, which allows us to deal with continuous, discrete, or statistical models in the same unified framework, and leads us to a broad class of numerical algorithms recently implemented in a Python library (namely, CodPy). Specifically, we propose a mesh-free discretization technique based on the theory of reproducing kernels and the theory of transport mappings, in a way that is reminiscent of Lagrangian methods in computational fluid dynamics. We introduce kernel-based discretizations of a variety of differential and discrete operators (gradient, divergence, Laplacian, Leray projection, extrapolation, interpolation, polar factorization). The proposed algorithms are nonlinear in nature and enjoy quantitative error estimates based on the notion of discrepancy error, which allows one to evaluate the relevance and accuracy of, both, the given data and the numerical solutions. Our strategy is relevant when a large number of degrees of freedom are present as is the case in mathematical finance and machine learning. We consider the Fokker-Planck-Kolmogorov system (relevant for problems arising in finance and material dynamics) and a class of neural networks based on support vector machines.

翻译：我们提出一种数值方法，称为基于传输的无网格方法，该方法能够在同一统一框架下处理连续、离散或统计模型，并引出一类最近在Python库（即CodPy）中实现的广泛数值算法。具体而言，我们基于再生核理论和传输映射理论提出一种无网格离散化技术，其思路类似于计算流体动力学中的拉格朗日方法。我们引入基于核的多种微分算子与离散算子（梯度、散度、拉普拉斯算子、Leray投影、外推、插值、极分解）的离散化形式。所提出的算法本质上具有非线性特性，并基于差异误差概念享有定量误差估计，从而允许评估给定数据与数值解的相关性与精确性。当存在大量自由度时（如数学金融与机器学习中的情形），我们的策略具有重要应用价值。我们考虑了Fokker-Planck-Kolmogorov系统（适用于金融与材料动力学问题）以及一类基于支持向量机的神经网络。