Data-driven discovery of the mechanism of systems described by partial differential equations

Turing patterns, a phenomenon introduced by mathematician and computer scientist Alan Turing, are intricate spatial patterns that emerge in reaction-diffusion systems, reflecting the dynamic interplay between chemical reactions and diffusion. These patterns, ranging from spots to stripes, offer physicists a captivating playground to explore the fundamental principles governing self-organization in complex systems. Studying Turing patterns not only unveils the underlying mechanisms of pattern formation but also provides valuable insights into the universal principles guiding the spontaneous emergence of order in nature. This master thesis explores the application of a sparse regression framework to analyze synthetic Turing patterns and experimental data from Drosophila embryos. Focusing on the Brussellator model, the algorithm aims to identify coefficients of reaction-diffusion equations based on stationary state and oscillation data. In the analysis of synthetic Turing patterns, we observe that the algorithm's success is intricately linked to parameter selection, particularly the regularization strength and sparsity threshold. The delicate balance between the model's error and complexity, as measured by interaction sparsity, and illustrated by a complexity-error tradeoff. Despite promising results in low to moderate noise scenarios, the algorithm's sensitivity to noise, especially in Laplacian computation, remains a limitation. Extending the study to synthetic oscillation data reveals the algorithm's improved robustness to noise, with successful identification of diffusion in the second-best sparse model. However, challenges persist in accurate diffusion constant estimation and sensitivity to noise (mainly in derivative calculations). Applying the sparse regression framework to Drosophila Turing patterns uncovers additional challenges, primarily related to the high noise level in Laplacian computation, limiting the algorithm's accuracy in identifying reaction terms. The conclusions highlight the need for future developments, including the exploration of noise-robust methods, data augmentation strategies, integration with biological models, and advanced parameter optimization techniques to enhance the framework's robustness and applicability to real-world cases. Our work has thus contributed to a deeper understanding of complex biological phenomena governed by reaction-diffusion dynamics.