The success of neural network–based machine learning methods stems from their ability to accurately approximate a wide range of functions, enabling them to effectively address diverse real-world problems. This capability is rooted in universal approximation theorems, which position neural networks as versatile and powerful tools for tasks such as image recognition, natural language processing, and predictive modeling.
However, achieving this requires a deep understanding of expressiveness, which determines how well a neural network can capture intricate patterns. This understanding is also key to addressing challenges like overfitting and ensuring robust generalization to unseen data. The study of neural network architectures is also important, including the design of their structures and the choice of activation functions. Developing efficient architectures is a key goal because it ensures models that balance computational cost with high performance across these applications.
Our research group has investigated how the norm of parameters of approximating neural networks behaves asymptotically (see [HM2024]). This is a crucial factor for establishing consistency results in training and plays a significant role in our convergence results for structured model learning, as well as in conducting a full error analysis. In other work, we derived approximation results up to the first derivative with rates for neural networks that use rational functions as activation functions (see [HM2025]).
Positron Emission Tomography (PET) is a non-invasive imaging technique that uses radioactive tracers to visualize and measure metabolic activity in the body. By detecting gamma rays from tracer interactions, PET creates detailed 3D images of biological processes. Dynamic PET goes a step further by capturing a series of images over time, providing both spatial and temporal insights into how tracers move and interact, revealing metabolic rates and mechanisms. However, PET imaging faces significant challenges: Reconstructiong a PET image from corresponding measurements requires to solve an inverse problem with a highly ill-posed forward model. Further, PET raw data is usually corrupted by particularly strong noise, which is even more pronounced in dynamic brain PET.
Our research group works on improving PET imaging in several aspects: In some of our works, such as [SHRVKBBN2017, SH2022, SHKBKBN2017, KHKOBS2017] we deal with modelling- and algorithmic aspects of PET image reconstruction, developing efficient algorithms and MR-prior based approaches to improve the quality of the reconstructed images. In other works, such as [HM2024, HM2025], we address more fundamental questions regarding the unique identifiability of physiological parameters in PET tracer kinetic modeling.
The field of learning PDE-based models from data is growing rapidly. It is driven by the need to bridge physical principles with data-driven insights and enables successful advances in learning unknown parameters in partially specified PDEs, discovering entirely new PDE structures, and approximating solution operators with remarkable accuracy. Alongside practical applications, there is increasing attention to theoretical analysis addressing consistency, convergence, and generalization. These are essential for ensuring the reliability and robustness of learned models in scientific and engineering contexts.
Our research group focuses on the framework of "structured model learning." Structured model learning builds on approximate physical knowledge and addresses the limitations of overly coarse abstractions by incorporating fine-scale physics learned from data. This approach improves interpretability, accuracy, and generality, enabling the model to handle complex scenarios and external factors beyond the scope of simplified physical principles. Augmenting known physics with data-driven components enables the model to effectively describe phenomena, even in non-ideal or challenging contexts. However, it is crucial to learn only what is necessary to ensure that the augmentation remains efficient and grounded in physical understanding.
Our research group is actively engaged in structured model learning, focusing on both the theoretical foundations and the practical applications. We have investigated the unique identifiability of learned fine-scale physics and established a convergence result with practical relevance (see [HM2024]). Currently, a major focus of our work is learning multi-pool dynamics in magnetic resonance imaging (MRI).
