面向复杂区域偏微分方程的几何感知隐式自回归生成模型

中文摘要 AI 辅助

求解多物理场偏微分方程（PDE）仍是科学计算中的重大挑战，尤其对于能量与化学工程中至关重要的微米级高度复杂曲折几何结构。为应对这一挑战，本文提出面向PDE的几何感知隐式自回归生成模型（GeoLAMP），用于求解高度不规则且曲折结构内的物理问题。GeoLAMP在图表示上引入双编码器架构，以联合捕捉全局拓扑与细尺度几何特征，实现从实空间场到紧凑隐式表示的有效转换；在隐式空间中，提出结合流匹配的因果自注意力Transformer以建模时间动态，支持稳定且可扩展的分块自回归预测；灵活的解码器可在任意点重建高分辨率物理场。本文构建了三个复杂几何下的多物理场基准数据集，涵盖反应流、热对流与弹性问题。GeoLAMP在这些数据集上始终实现最稳定的自回归性能，在整个推演过程中保持低误差。研究为微米级复杂几何下PDE的几何感知学习提供了系统研究，并通过流匹配框架为隐式自回归PDE建模的分块时间推进提供了新见解。

英文摘要

Solving multiphysics partial differential equations (PDEs) remains a major challenge in scientific computing, especially for highly complex $μ$m-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 $μ$m-scale complex geometries and offer new insights into block-wise time marching of latent autoregressive PDE modeling via a flow matching framework.

发表机构

• Stanford University（斯坦福大学）

机构由 AI 辅助整理，请以论文原文为准。