Model selection focusing on longtime behavior of differential equations

Modeling biological processes is a highly demanding task because not all processes are fully understood. Mathematical models allow us to test hypotheses about possible mechanisms of biological processes. The mathematical mechanisms oftentimes abstract from the biological micro-scale mechanisms. Experimental parameter calibration is extremely challenging as the connection between abstract and micro-scale mechanisms is unknown. Even if some microscopic parameters can be determined by isolated experiments, the connection to the abstract mathematical model is challenging. We present ideas for overcoming these difficulties by using longtime characteristics of solutions for, first, finding abstract mechanisms covering large-scale observations and, second, determining parameter values for the abstract mechanisms. The parameter values are not directly connected to experimental data but serve as a link between known mechanisms and observations. The framework combines machine learning techniques with the characteristic solution behavior of differential equations. This setting gives insight into challenges by using rare data only that can later be used for partial differential equations.