Geometry-aware Latent Autoregressive Generative Model for PDEs in Complex Domains

Solving multiphysics partial differential equations (PDEs) remains a major challenge in scientific computing, especially for highly complex μm-scale tortuous geometries critical to energy and chemical engineering. We address this challenge by proposing a Geometry-aware Latent Autoregressive generative Model for PDEs (GeoLAMP), which solves physics within highly irregular and tortuous structures by decoupling flow and transport physics. GeoLAMP introduces a dual-encoder architecture on graph representations to jointly capture global topology and fine-scale geometric features, enabling an effective transition from real-space fields to compact latent representations. In the latent space, we propose a causal self-attention transformer with flow matching to model temporal dynamics, allowing stable and scalable block-wise autoregressive prediction. In addition, we propose a grid-graph data fusion scheme that projects low-resolution grid-based approximate priors onto graph representations, improving prediction of flow in tortuous structures. We establish three multiphysics benchmark datasets in complex geometries, covering reactive flow, heat convection, and elasticity. GeoLAMP consistently achieves the most stable autoregression performance on these datasets. Our results provide a systematic study of geometry-aware learning for PDEs in μm-scale complex geometries and offer new insights into block-wise time marching of latent autoregressive PDE modeling via a flow matching framework.