Project Details
Projekt
Kolmogorov-Arnold-Wiener Finite Volume Neural Network: Learning Physics in Porous Media from Experimental Data
Subject Area
Hydrogeology, Hydrology, Limnology, Urban Water Management, Water Chemistry, Integrated Water Resources Management
Term
since 2026
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 576425327
Understanding and predicting how complex physical processes behave in the real world, for example, how pollutants move through groundwater, is critical for solving environmental, industrial, and societal challenges. Traditionally, scientists rely on mathematical equations in the form of partial differential equations (PDEs) to describe these processes. However, such systems are often only partially understood; therefore, the equations are only partially known or hard to apply. In recent years, machine learning (ML) has opened up new opportunities in this area. ML models can “learn” physical behaviour from data, offering an alternative when classical PDEs fall short. However, today’s ML models are predominantly data-hungry and difficult to interpret. More importantly, conventional ML approaches ignore even those parts of physical laws that are already known, such as conservation of mass, energy, or momentum, making them less reliable for use in scientific applications. Recent progress has combined physics-based modelling with machine learning. This hybrid approach brings together the logic and robustness of physics with the flexibility and learning ability of ML. Our project builds on this foundation idea and takes it to a new level: we aim to develop a hybrid approach as a tool to support the scientific investigation of poorly understood phenomena in complex systems. To achieve this, we will develop a novel neural network based on the Kolmogorov-Arnold and Wiener theorems (KAW-NN). Then, we will combine it with the Finite Volume Neural Network (FINN), a physics-aware method for machine learning to ensure that conservation laws are respected. Our combined approach, resulting in the Kolmogorov-Arnold-Wiener Finite Volume Neural Network (KAW-FINN), will integrate four key areas: (1) PDE-based modeling, (2) numerical simulation, (3) Kolmogorov-Arnold networks, and (4) Wiener networks. This unique combination will stand out due to its compact structure, reduced hunger for data, mathematically grounded interpretability, and ability to efficiently represent highly nonlinear relationships. Additionally, we will include uncertainty quantification to support more robust scientific analysis, hypothesis testing, and decision-making. We expect the KAW-FINN framework to deliver unprecedented accuracy and interpretability when investigating complex physical systems. As demonstrator applications, we focus on non-linear transport and reaction processes in porous media, including variably saturated, non-Fickian solute transport. Overall, we will develop a novel tool to better support the scientific investigation of complex systems, and also help understand, model, and predict poorly understood phenomena from measurement data to close important scientific knowledge gaps.
DFG Programme
Research Grants
International Connection
USA
Cooperation Partner
Professor Daniel Tartakovsky, Ph.D.
