Modeling Dynamics over Meshes with Gauge Equivariant Nonlinear Message Passing
This work addresses a domain-specific problem in computer graphics and physical systems for researchers and practitioners dealing with PDEs on non-Euclidean manifolds, though it is incremental as it builds on existing gauge equivariant methods.
The paper tackled the problem of modeling partial differential equations (PDEs) on surface meshes by introducing a gauge equivariant nonlinear message passing architecture, which achieved higher performance than convolutional or attentional networks on domains with complex nonlinear dynamics.
Data over non-Euclidean manifolds, often discretized as surface meshes, naturally arise in computer graphics and biological and physical systems. In particular, solutions to partial differential equations (PDEs) over manifolds depend critically on the underlying geometry. While graph neural networks have been successfully applied to PDEs, they do not incorporate surface geometry and do not consider local gauge symmetries of the manifold. Alternatively, recent works on gauge equivariant convolutional and attentional architectures on meshes leverage the underlying geometry but underperform in modeling surface PDEs with complex nonlinear dynamics. To address these issues, we introduce a new gauge equivariant architecture using nonlinear message passing. Our novel architecture achieves higher performance than either convolutional or attentional networks on domains with highly complex and nonlinear dynamics. However, similar to the non-mesh case, design trade-offs favor convolutional, attentional, or message passing networks for different tasks; we investigate in which circumstances our message passing method provides the most benefit.