AROMA: Preserving Spatial Structure for Latent PDE Modeling with Local Neural Fields
This work addresses efficient and stable simulation of PDEs for applications in physics and engineering, representing an incremental improvement with a novel hybrid method.
The paper tackles modeling partial differential equations (PDEs) by introducing AROMA, a framework using local neural fields to preserve spatial structure, which achieves greater stability and enables longer rollouts compared to conventional methods.
We present AROMA (Attentive Reduced Order Model with Attention), a framework designed to enhance the modeling of partial differential equations (PDEs) using local neural fields. Our flexible encoder-decoder architecture can obtain smooth latent representations of spatial physical fields from a variety of data types, including irregular-grid inputs and point clouds. This versatility eliminates the need for patching and allows efficient processing of diverse geometries. The sequential nature of our latent representation can be interpreted spatially and permits the use of a conditional transformer for modeling the temporal dynamics of PDEs. By employing a diffusion-based formulation, we achieve greater stability and enable longer rollouts compared to conventional MSE training. AROMA's superior performance in simulating 1D and 2D equations underscores the efficacy of our approach in capturing complex dynamical behaviors.