Empirical Approach on Uncertainty Analysis of Differential Equations in Mathematical Modeling

Abstract

Uncertainty theory, introduced by Professor Baoding Liu of Tsinghua University in China in 2007, has lately made significant progress. It serves as an advanced and adaptable mathematical tool for modeling uncertainties and handling unforeseen outcomes that may arise when using likelihood and/or fuzzy set approaches, which are sometimes favored alternatively. Mathematical models are used in several academic disciplines to provide accurate quantitative estimations based on facts. Crucially, these mathematical frameworks must offer a dependable evaluation of the certainty in their predictions. Uncertainty quantification (UQ) is a discipline that focuses on delivering accurate and dependable assessments of trust regarding forecasts made by models. Empirical models utilize data and statistical methods to build connections between parameters in a system, whereas mechanistic frameworks are built upon prior knowledge and understanding of the underlying mechanism that governs changes in the overall structure. This article will demonstrate the effectiveness of a data-driven empirical technique in solving ordinary differential equations under conditions of uncertainty. The research aims to determine the practicality and accuracy of its approach compared to existing numerical methods, which may often yield unsatisfactory results.