Abstract
Graphs are widely used to model interconnected systems, offering powerful tools for data representation and problem-solving. However, their reliance on pairwise, single-type, and static connections limits their expressive capacity. Recent developments extend this foundation through higher-order structures, such as hypergraphs, multilayer, and temporal networks, which better capture complex real-world interactions. Many real-world systems, ranging from brain connectivity and genetic pathways to socio-economic networks, exhibit multimodal and higher-order dependencies that traditional networks fail to represent. This paper introduces a novel generalisation of message passing into learning-based function approximation, namely multimodal heterogeneous network flow, which models information propagation across different semantic domains under conservation constraints. This framework is defined via Petri nets, which extend hypergraphs to support concurrent, multimodal flow and richer structural representation. Building on this foundation, we present the Petri Graph Neural Network (PGNN), a new class of graph neural networks capable of learning over higher-order, multimodal structures. PGNN generalises message passing by incorporating flow conversion and concurrency, leading to enhanced expressive power, interpretability, and computational efficiency. The work opens new directions in learning over complex structures, transcending transformer-based and traditional hypergraph-based algorithms. We validate results through theoretical analysis and real-world experiments, while demonstrating superior performance in, e.g., stock market prediction.