Physics-Informed DeepONets for drift-diffusion on metric graphs: simulation and parameter identification

We develop a novel physics informed deep learning approach for solving nonlinear drift-diffusion equations on metric graphs. These models represent an important model class with a large number of applications in areas ranging from transport in biological cells to the motion of human crowds. While traditional numerical schemes require a large amount of tailoring, especially in the case of model design or parameter identification problems, physics informed deep operator networks (DeepONets) have emerged as a versatile tool for the solution of partial differential equations with the particular advantage that they easily incorporate parameter identification questions. We here present an approach where we first learn three DeepONet models for representative inflow, inner and outflow edges, resp., and then subsequently couple these models for the solution of the drift-diffusion metric graph problem by relying on an edge-based domain decomposition approach. We illustrate that our framework is applicable for the accurate evaluation of graph-coupled physics models and is well suited for solving optimization or inverse problems on these coupled networks.

The simulation of flows in networks is a challenging task. We have developed a new AI-based method to help solve such complex physical models that describe how objects like particles or people move through connected pathways—similar to networks or graphs. These models are useful in many real-world situations, such as understanding how substances move within cells or how people flow through crowded areas. Our approach uses a modern form of machine learning called physics-informed deep operator networks (DeepONets). We trained separate AI models for different parts of the network (such as where things enter, move through, or exit). Then, we connected these models together in a way that respects the structure of the whole network. Our results show that this method can accurately simulate these types of systems and can also be used to solve related challenges, like optimizing the system or identifying unknown parameters.