arXiv subjects

Josef Daněk

Publications and source records attributed to Josef Daněk.

2 recordsLinked to original sources

arXiv2025

Challenges in automatic differentiation and numerical integration in physics-informed neural networks modelling

In this paper, we numerically examine the precision challenges that emerge in automatic differentiation and numerical integration in various tasks now tackled by physics-informed neural networks (PINNs). Specifically, we illustrate how ill-posed problems or inaccurately computed functions can cause serious precision issues in differentiation and integration. A major difficulty lies in detecting these problems. A simple large-scale view of the function or good-looking loss functions or convergence results may not reveal any potential errors, and the resulting outcomes are often mistakenly considered correct. To address this, it is critical to determine whether standard double-precision arithmetic suffices or if higher precision is necessary. Three problematic use-cases for solving differential equations using PINNs are analysed in detail. For the case requiring numerical integration, we also evaluate several numerical quadrature methods and suggest particular numerical analysis steps to choose the most suitable method.

math.NA↗

arXiv2020

Numerical aspects of integration in semi-closed option pricing formulas for stochastic volatility jump diffusion models

In mathematical finance, a process of calibrating stochastic volatility (SV) option pricing models to real market data involves a numerical calculation of integrals that depend on several model parameters. This optimization task consists of large number of integral evaluations with high precision and low computational time requirements. However, for some model parameters, many numerical quadrature algorithms fail to meet these requirements. We can observe an enormous increase in function evaluations, serious precision problems and a significant increase of computational time. In this paper we numerically analyse these problems and show that they are especially caused by inaccurately evaluated integrands. We propose a fast regime switching algorithm that tells if it is sufficient to evaluate the integrand in standard double arithmetic or if a higher precision arithmetic has to be used. We compare and recommend numerical quadratures for typical SV models and different parameter values, especially for problematic cases.

math.NA↗