PANO: Parallel Latent Space Neural Operator for Solving PDEs

Aggregating observations into a limited number of latent tokens and learning physical interactions in the latent space has emerged as an effective approach for Transformer-based neural operators to solve Partial Differential Equation (PDE) problems involving large-scale observations. However, existing methods typically compress observations using a single aggregation pattern, making physical information not retained by that pattern invisible to the latent space and thereby constraining physical interactions within it. To address this limitation, we propose PArallel Latent Space Neural Operator (PANO), a novel neural operator for solving PDEs that formulates physical interactions among observations as an additive combination of interaction results from parallel latent spaces with shared parameters. Specifically, PANO transforms observations into multiple distinct compressed representations through independent mappings, learns physical interactions in parallel based on these representations, and finally fuses the decoded results to predict the target physical field. Within PANO, we further propose Coupled Assignment Projection (CAP) to construct bidirectionally consistent encoding and decoding mappings, and Pairwise Latent Space Attention Layer (PaLA) to jointly update latent representations of each latent space through adaptive cross-space interactions. Extensive experiments on nine benchmarks spanning diverse physical systems, geometries, and observation scales demonstrate that PANO achieves state-of-the-art performance. Notably, PANO reduces equivalent plastic strain prediction errors on Unilateral Stamping by 28.3% (or more) and lift-coefficient prediction errors on AirCraft by 21.4% (or more) compared with existing methods. The source code will be released.