Modeling complex systems with differential equations
type: CreativeWork
description: Mathematical models have since long been successful in describing nature and specifically dynamical processes of real-world systems. Solely relying on mathematical formalism, it has become possible to make adequate predictions of the temporal evolution of systems of all kind and furthermore to control processes from outside. However, despite the fact that mathematical models are more and more able to describe processes on smallest and largest scales and theories unify, it is not reasonable to try to describe all processes with one formalism. On the contrary, mathematical models seem to be confined to different levels of complexity since mathematical approaches that work for small scales are not manageable in systems with increasing complexity. For example, quantum mechanics is well suited for small scales, however for describing the temporal evolution of macroscopic systems, the quantum mechanical ansatz is not applicable not to speak of even more complex systems. Similar to statistical mechanics, respectively thermodynamics, one is not interested in the behavior of the wave function of every atom but in variables defining the system state on larger scales. Departing from first principles and modeling mesoscopic or macroscopic systems with 'appropriate' variables, often leads to the situation where, for one system to be modeled, different mathematical descriptions arise which are motivated from first principles. One then faces the situation where it is a priori unclear which mathematical model is best suited to describe the system state and its temporal evolution. Additionally, through the approximative nature, these mathematical models often contain unknown parameters which cannot be derived from universal constants. This leads to the so-called inverse problem where it is necessary to estimate unknown parameters with help of experimental data. Beforehand it is additionally necessary to analyze identifiability of candidate models.
abstract: Mathematical models have since long been successful in describing nature and specifically dynamical processes of real-world systems. Solely relying on mathematical formalism, it has become possible to make adequate predictions of the temporal evolution of systems of all kind and furthermore to control processes from outside. However, despite the fact that mathematical models are more and more able to describe processes on smallest and largest scales and theories unify, it is not reasonable to try to describe all processes with one formalism. On the contrary, mathematical models seem to be confined to different levels of complexity since mathematical approaches that work for small scales are not manageable in systems with increasing complexity. For example, quantum mechanics is well suited for small scales, however for describing the temporal evolution of macroscopic systems, the quantum mechanical ansatz is not applicable not to speak of even more complex systems. Similar to statistical mechanics, respectively thermodynamics, one is not interested in the behavior of the wave function of every atom but in variables defining the system state on larger scales. Departing from first principles and modeling mesoscopic or macroscopic systems with 'appropriate' variables, often leads to the situation where, for one system to be modeled, different mathematical descriptions arise which are motivated from first principles. One then faces the situation where it is a priori unclear which mathematical model is best suited to describe the system state and its temporal evolution. Additionally, through the approximative nature, these mathematical models often contain unknown parameters which cannot be derived from universal constants. This leads to the so-called inverse problem where it is necessary to estimate unknown parameters with help of experimental data. Beforehand it is additionally necessary to analyze identifiability of candidate models.
author: Müller, Thorsten
keywords: Modellselektion, Parameterschätzung, Identifizierbarkeit, Model selection, data analysis, dynamical systems, parameter estimation, identifiability
date modified: 2026-07-24T02:21:46Z
provider: University of Freiburg
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Freiburg