AROMA: Preserving Spatial Structure for Latent PDE Modeling with Local Neural Fields

Louis Serrano, Thomas X Wang, Etienne Le Naour, Jean-Noël Vittaut, Patrick Gallinari

Keywords: PDE, Neural Operator, Neural Fields, Transformer, Diffusion

TL;DR: We introduce AROMA, a versatile framework for enhanced PDE modeling using local neural fields, which achieves stable, efficient processing of diverse spatial data and superior performance in simulating 1D and 2D equations.

Abstract: 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.

Submission Number: 232