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.

翻译：建模生物过程是一项高度要求性的任务，因为并非所有过程都被完全理解。数学模型使我们能够检验关于生物过程可能机制的假设。数学机制通常抽象于生物微观尺度机制。实验参数校准极具挑战性，因为抽象机制与微观尺度机制之间的联系未知。即使某些微观参数可以通过孤立实验确定，其与抽象数学模型之间的关联仍很困难。我们提出了克服这些难题的思路，通过利用解的长期特征：首先寻找涵盖大规模观测的抽象机制，其次确定抽象机制的参数值。参数值不直接与实验数据关联，但可作为已知机制与观测之间的桥梁。该框架将机器学习技术与微分方程的特征解行为相结合。这一设置揭示了仅使用稀疏数据的挑战，这些方法后续可应用于偏微分方程。