Data-Driven Modeling and Parameter Estimation for Reaction-Diffusion Systems

Abstract

One aspect of this thesis is the exploration of data-driven model reduction techniques for efficiently analyzing the dynamic behavior captured in temporal datasets generated by reaction-diffusion partial differential equation systems. The first such technique is the Dynamic Mode Decomposition, an equation-free method originally introduced by Peter Schmid and Jörn Sesterhenn in 2008 ([23]). Building on this, a randomized variant of Dynamic Mode Decomposition is employed to improve computational efficiency. However, both versions struggle in providing accurate reconstructions for datasets that exhibit periodic behavior or spatio-temporal Turing instability. To address this issue, we propose a piecewise approach that partitions the datasets and applies Dynamic Mode Decomposition locally to each subset. The second technique is Proper Orthogonal Decomposition, which is a well-established method for model order reduction. To further reduce computational costs, Proper Orthogonal Decomposition is combined with the Discrete Empirical Interpolation Method. Despite this, the reconstruction accuracy remains insufficient in some cases. Therefore, we introduce a correction-based strategy to enhance the quality of the reduced model. Moreover, by leveraging the specific structure of datasets that exhibit Turing instability, we improve the computational effectiveness even further by extending the previous approaches in an adaptive manner. Another key aspect of this thesis is parameter identification. The proposed strategy relies on computing gradients of the cost functional using a sensitivity approach. These gradients are then used within the projected Barzilai-Borwein optimization method to identify optimal parameter values. Finally, we investigate a specific reaction-diffusion system in which nonlinearities in the reaction kinetics arise from a Hill function, commonly used to model cooperative effects in biochemical reactions.