Hybrid Lagrangian-Eulerian Model for Lagrangian Fluid Simulation

Fluid dynamics are typically discretized using either an Eulerian formulation, which resolves physical fields on fixed spatial nodes, or a Lagrangian formulation, which tracks discrete moving particles carrying physical quantities along their trajectories. Building on the latter, Lagrangian neural simulators learn physical dynamics by predicting particle state transitions over time, offering exceptional geometric flexibility and exact advection for modeling free-surface fluids and moving boundaries. However, the absence of a fixed global reference frame as used in Eulerian formulation introduces two severe limitations: (1) spatial redundancy, as accurate gradients require densely packed particles, many of which move nearly identically and waste model capacity; and (2) temporal error accumulation, as small trajectory errors corrupt the local neighborhoods used for information exchange and compound over rollouts. Inspired by classical hybrid numerical solvers, we propose a Hybrid Lagrangian–Eulerian neural solver that augments Lagrangian dynamics with an Eulerian representation. To reduce spatial redundancy, an adaptive downsampler learns to prune particles that move nearly identically to their neighbors, focusing model capacity on regions with complex dynamics; the retained particles are then aggregated onto Eulerian nodes to resolve large-scale dynamics. To counter temporal error accumulation, cross-attention lets the aggregated particle features query an Eulerian representation on fixed nodes, which serves as a stable spatial anchor to correct trajectory deviations at every timestep. We conducted comprehensive experiments across 16 datasets and 16 baselines, establishing the most extensive benchmark in the Lagrangian neural simulation literature. Our results demonstrate that this approach establishes a new state-of-the-art.