Topology-aware Neural Flux Prediction Guided by Physics

Graph Neural Networks (GNNs) often struggle in preserving high-frequency components of nodal signals when dealing with directed graphs. Such components are crucial for modeling flow dynamics, without which a traditional GNN tends to treat a graph with forward and reverse topologies equal. To make GNNs sensitive to those high-frequency components thereby being capable to capture detailed topological differences, this paper proposes a novel framework that combines 1) explicit difference matrices that model directional gradients and 2) implicit physical constraints that enforce messages passing within GNNs to be consistent with natural laws. Evaluations on two real-world directed graph data, namely, water flux network and urban traffic flow network, demonstrate the effectiveness of our proposal.

Predicting how physical quantities like energy or material flow (a.k.a. flux) behave in complex systems is critical in fields such as environmental science, fluid dynamics, and ecological engineering. Physics-based simulators, while built upon first principles, can be computationally expensive and difficult to scale or adapt to dynamic, real-time scenarios. In contrast, data-driven approaches like machine learning offer flexibility and efficiency, yet they often ignore physical constraints and may yield predictions that violate fundamental laws of physics -- for instance, erroneously predicting flux propagation from downstream to upstream, which contradicts the natural water flow direction.This paper presents a physics-guided framework that integrates graph neural networks (GNNs) with governing physical laws to improve neural flux prediction. Rather than treating GNNs as black boxes, our method encodes conservation laws directly into the message-passing process, ensuring that learned representations are physically consistent. The resultant model PhyNFP enhances directional distinguishability, exhibits robustness to abrupt perturbations, and yields physically interpretable outputs. These merits collectively lend PhyNFP a reliable and trustworthy tool for real-time forecasting and decision-making in high-stakes settings such as flood forecasting and environmental risk management.