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A Class of Mesh-Free Algorithms for Some Problems Arising in Finance and Machine Learning

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Abstract

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.

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Enquiries about data availability should be directed to the authors.

Notes

  1. By consistency, we mean that the schemes can only converge to a solution to the given problem.

  2. We do not perform here comparisons with existing algorithms (a task tackled in [19]).

  3. The points are arbitrary at this stage while later they will be chosen in order to minimize an error functional such as (2.2) below.

  4. In practice, the error estimate should also take into account the are approximation errors, in the form of a variance term taking into account the variability of random data.

  5. This expression is motivated from (2.8) and (2.9).

  6. Here, the term “extrapolation” is used when some data defined on a “small” set are extended to a “large” set. Statistician reader might prefer a different terminology here.

  7. Our notation may be unconventional for a statistician reader, as we use Y for a regression variable (and not for a dependent variable).

  8. As define at https://www.tensorflow.org/tutorials/quickstart/beginner.

  9. By definition, i.i.d. variables are independent and identically distributed random variables.

  10. For convex maps, the choice \(\partial _t T^{-1} = \zeta \circ T^{-1}\) is an alternate and simpler computational choice.

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, Centre National de la Recherche Scientifique, Sorbonne Université, 4 Place Jussieu, 75252, Paris, France

    Philippe G. LeFloch

  2. MPG-Partners, 136 Boulevard Haussmann, 75008, Paris, France

    Jean-Marc Mercier

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  1. Philippe G. LeFloch
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Correspondence to Philippe G. LeFloch.

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LeFloch, P.G., Mercier, JM. A Class of Mesh-Free Algorithms for Some Problems Arising in Finance and Machine Learning. J Sci Comput 95, 75 (2023). https://doi.org/10.1007/s10915-023-02179-5

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