ATTENTION MECHANISM FOR SCALABLE MESH-BASED NEURAL SURROGATES OF FREE-SURFACE FLUIDS

High-fidelity simulations of free-surface flows, particularly those involving non-Newtonian rheologies, are computationally demanding due to the need for continuous domain updates and the resolution of material nonlinearities. The Particle Finite Element Method (PFEM) is a standard Lagrangian approach for these problems, but it remains expensive. While graph neural networks (GNNs) have been used as surrogates, they suffer from memory bottlenecks and scalability issues because they rely on local message-passing operations that require multiple steps to propagate information across the domain.

To address this, the authors propose a mesh-based neural surrogate that replaces traditional GNN message-passing with attention mechanisms. By using either standard or linear attention, the model learns spatial correlations adaptively without needing to store explicit edge features or perform multiple neighborhood-based message-passing iterations. This architecture preserves the PFEM mesh discretization, allowing for consistent remeshing and node redistribution, which are critical for maintaining stability during long-term simulations.

The study demonstrates that attention-based surrogates can accurately predict both transient dynamics and final configurations of free-surface flows. By eliminating the need for explicit edge embeddings, the framework significantly reduces memory overhead and improves scalability for large-scale 2D and 3D problems. The authors show that the linear attention variant, in particular, provides a scalable solution for complex engineering-scale applications, including non-Newtonian fluids where material properties vary.

This work provides a robust, scalable alternative to traditional GNN-based surrogates in Lagrangian fluid mechanics. By integrating deep learning with the geometric and topological framework of PFEM, the proposed method enables the simulation of complex, evolving geometries that were previously too computationally expensive to model with high-fidelity surrogates. The ability to reconstruct mechanical quantities like stress fields directly from the surrogate output further enhances its utility for practical engineering design.