Modeling complex systems with differential equations

Mathematical models have since long been successful in describing nature <br>and specifically dynamical processes of real-world systems. <br>Solely relying on mathematical <br>formalism, it has become possible to make adequate predictions of <br>the temporal evolution of systems of all kind and furthermore to control <br>processes from outside. <br>However, despite the fact that mathematical models are more and more able <br>to describe processes on smallest and largest scales and theories <br>unify, it is not reasonable to try to describe all processes with one <br>formalism. On the contrary, mathematical models seem to <br>be confined to different levels of complexity since mathematical <br>approaches that work for small scales are not manageable in systems with <br>increasing complexity. <br>For example, quantum mechanics is well suited <br>for small scales, <br>however for describing the temporal evolution of macroscopic systems, the <br>quantum mechanical ansatz is not applicable not to <br>speak of even more complex systems. Similar to statistical mechanics, <br>respectively thermodynamics, <br>one is not interested in the behavior of the wave function of every <br>atom but in variables defining the system state on larger scales. <br>Departing from first principles and modeling mesoscopic or <br>macroscopic systems with 'appropriate' variables, often leads to the <br>situation where, for one system to be modeled, different mathematical <br>descriptions arise which are motivated from <br>first principles. One <br>then faces the situation where it is a priori unclear which <br>mathematical model is best suited to describe the system state and its <br>temporal evolution. <br>Additionally, through the approximative nature, these mathematical <br>models often contain unknown parameters <br>which cannot be derived from universal constants. This leads to the <br>so-called inverse problem where it is necessary to estimate unknown <br>parameters with help of experimental data. Beforehand it is additionally necessary to analyze identifiability of candidate models.

Author dc:creator

• Müller, Thorsten

Contributors dc:contributor

• Honerkamp, Josef

• Modellselektion
• Parameterschätzung
• Identifizierbarkeit
• Model selection
• data analysis
• dynamical systems
• parameter estimation
• identifiability

Repository record source_url

https://freidok.uni-freiburg.de/data/556

OAI identifier oai:identifier

oai:freidok.uni-freiburg.de:556