Conservative Local-Global Transport: Mesh-Local Convolution and Scalable Linear Attention for Neural Lagrangian Simulation

Many neural Lagrangian simulators rely on Graph Neural Networks (GNNs) with spatially constructed interaction graphs, introducing considerable topology-processing and message-passing costs as the particle count grows. Moreover, these architectures are typically agnostic to the underlying conservation laws. In this work, we propose a conservative local–global transport mechanism in latent space: mesh convolution captures strongly coupled near-field interactions, while separable linear attention enables dense long-range interactions at linear complexity, both enforcing conservation laws by construction. Beyond global conservation, we introduce sign-split latent flux limiters that strictly preserve prescribed physical bounds without breaking the conservative latent update, and connect these discrete limiters to a unified family of latent advection dynamics. It demonstrates advantages over GNN-based baselines in computational efficiency (supports up to $38\times$ more particles) and physical consistency while maintaining competitive predictive accuracy. Our code is available at https://anonymous.4open.science/r/CONSERVATIVE-LOCAL-GLOBAL-TRANSPORT-I16S.