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).
