几何与物理的融合：面向非结构化神经偏微分方程求解器的数据高效预训练
Geometry Meets Physics: Data-Efficient Pre-Training for Unstructured Neural PDE Solvers
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中文总结 AI 辅助
针对非结构化三维PDE神经求解器，提出几何与物理驱动的无磁盘数据预训练框架，提升低数据场景下的收敛速度、数据效率与精度。
中文摘要 AI 辅助
在非结构化三维几何上的偏微分方程（PDE）神经代理模型，常常受限于泛化能力差以及生成大规模训练数据集的高昂成本。因此，在相关PDE动力学的大规模数据集上进行预训练，已成为增强这些模型鲁棒性和可扩展性的关键替代方案。然而，该策略在计算和数据两方面均非高效，因为它依赖于大规模预计算数据，而这些数据的生成成本极高。在本工作中，我们引入了一种无磁盘数据的预训练框架，同时适用于稳态和瞬态两种情形。对于稳态问题，我们提出了一种几何驱动的策略，利用固有形状描述符来学习复杂三维域的表征。对于瞬态问题，我们提出了一种基于在线生成合成PDE数据的物理驱动方法，从而无需依赖昂贵数据集即可实现可扩展的预训练。在多项实验中，我们的方法在微调阶段实现了更快的收敛、更高的数据效率和更高的精度，尤其是在现实中的低数据场景下。该方法为大规模模拟中的数据高效神经仿真器提供了一条实用路径。
英文摘要
Neural surrogate models for Partial Differential Equations (PDEs) on unstructured 3D geometries are often limited by poor generalization and the high cost of generating large-scale training datasets. Consequently, pre-training on massive datasets of related PDE dynamics has emerged as a critical alternative to enhance the robustness and scalability of these models. However, this strategy is neither compute- nor data-efficient, as it relies on massive pre-computed data that is very costly to generate. In this work, we introduce a disk-data-free pre-training framework tailored to both steady-state and transient regimes. For steady-state problems, we propose a geometry-driven strategy that leverages intrinsic shape descriptors to learn representations of complex 3D domains. For transient problems, we introduce a physics-driven approach based on online generation of synthetic PDE data, enabling scalable pre-training without reliance on expensive datasets. Across multiple experiments, our approach achieves faster convergence, greater data efficiency, and higher accuracy during fine-tuning, particularly under realistic low-data regimes. This methodology provides a practical pathway toward data-efficient neural emulators for large-scale simulations.
发表机构
- Technical University of Munich（慕尼黑工业大学）
- University of Stuttgart（斯图加特大学）
机构由 AI 辅助整理，请以论文原文为准。
