Mathematical Modeling with ordinary and partial differential equations, qualitative behavior of solutions, spatial inhomogeneous reaction diffusion equations.
Applications in life sciences, engineering and environmental sciences.
Towards Model Discovery Using Domain Decomposition and PINNs Tirtho S. Saha, Alexander Heinlein, Cordula Reisch IFAC Papersonline, 2025 We enhance machine learning algorithms for learning model parameters in complex systems represented by differential equations with domain decomposition methods. The study evaluates the performance of two approaches, namely (vanilla) Physics-Informed Neural Networks (PINNs) and Finite Basis Physics-Informed Neural Networks (FBPINNs), in learning the dynamics of test models with a quasi-stationary longtime behavior. We test the approaches for data sets in different dynamical regions and with varying noise level. As results, the FBPINN approach better captures the overall dynamical behavior compared to the vanilla PINN approach, even in cases with data only from a time domain with quasi-stationary dynamics.
Impact of topography and combustion functions on fire front propagation in an advection-diffusion-reaction model for wildfires Luca Nieding, Cordula Reisch, Dirk Langemann, Adrián Navas-Montilla IFAC Papersonline, 2025 Given the recent increase in wildfires, developing a better understanding of their dynamics is crucial. For this purpose, the advection-diffusion-reaction model has been widely used to study wildfire dynamics. In this study, we introduce the previously unconsidered influence of topography through an additional advective term. Furthermore, we propose a linear term for the combustion function, comparing it with the commonly used Arrhenius law to offer a simpler model for further analysis. Our findings on the model’s dynamics are supported by numerical simulations showing the differences of model extensions and approximations.
Building up a model family for inflammations Cordula Reisch, Sandra Nickel, Hans-Michael Tautenhahn Journal of Mathematical Biology, 2024 The paper presents an approach for overcoming modeling problems of typical life science applications with partly unknown mechanisms and lacking quantitative data: A model family of reaction–diffusion equations is built up on a mesoscopic scale and uses classes of feasible functions for reaction and taxis terms. The classes are found by translating biological knowledge into mathematical conditions and the analysis of the models further constrains the classes. Numerical simulations allow comparing single models out of the model family with available qualitative information on the solutions from observations. The method provides insight into a hierarchical order of the mechanisms. The method is applied to the clinics for liver inflammation such as metabolic dysfunction-associated steatohepatitis or viral hepatitis where reasons for the chronification of disease are still unclear and time- and space-dependent data is unavailable.
Analytical and numerical insights into wildfire dynamics: Exploring the advection–diffusion–reaction model Cordula Reisch, Adrián Navas-Montilla, Ilhan Özgen-Xian Computers and Mathematics with Applications, 2024 Understanding the dynamics of wildfire is crucial for developing management and intervention strategies. Mathematical and computational models can be used to improve our understanding of wildfire processes and dynamics. This paper presents a systematic study of a widely used advection–diffusion–reaction wildfire model with non-linear coupling. The importance of single mechanisms is discovered by analysing hierarchical sub-models. Numerical simulations provide further insight into the dynamics. As a result, the influence of wind and model parameters such as the bulk density or the heating value on the wildfire propagation speed and the remaining biomass after the burn are assessed. Linearisation techniques for a reduced model provide surprisingly good estimates for the propagation speed in the full model.
