Partial Differential Equation Discovery Considering Differential Consequences

• Applicants: Professor Dr. Markus Lange-Hegermann; Professor Dr. Daniel Robertz
• Subject Area: Mathematics ; Methods in Artificial Intelligence and Machine Learning
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 576384230

Scientific research traditionally relies on observations to deduce laws, frequently expressed through differential equations. However, while such equations are common in mathematically underpinned disciplines, such as in physics, their prevalence in other sciences is limited. The increasing availability of data opens up opportunities to uncover additional laws, particularly in mathematically less explored scientific domains. The existing methods to date for partial differential equation discovery from data are restricted to specific types of equations, as they neglect differential consequences. This project seeks to pioneer a novel approach to identify partial differential equations from finite data sets, explicitly taking into account differential consequences. The proposed methodology integrates symbolic computation with data-driven techniques, employing existing equation discovery, stochastic processes, and computational differential algebra. The novel iterative process integrates existing equation discovery to suggest differential equations, utilizes the Thomas algorithm to incorporate differential consequences, and employs Gaussian processes to evaluate the fit of the proposed differential equations to the dataset. The overarching goals include developing high-quality open-source research software and validating the methods on real-world applications. Particular attention will be paid to accuracy, computational speed, data requirements, and interpretability. This research represents a crucial step towards creating interpretable machine learning algorithms that automatically extract human-understandable laws from data. Furthermore, the project bridges the gap between symbolic computation, algebra, pure mathematics, and data-driven methods, fostering collaboration between traditionally distant disciplines.

• DFG Programme: Research Grants