用可复用的局部模型拼接大域解，无需知道方程细节
Latent Space Element Method
- 局部子模型在隐空间学习，通过方向性交互连接
- 1D波方程测试中，训练域外扩展仍保持高精度
- 适合构建可解释、可扩展的通用代理求解器
如何构建在小域训练却能扩展到大域的代理求解器，且无需访问微分方程算子？受数据驱动有限元法（DD-FEM）框架启发，我们提出隐空间单元法（LSEM），一种基于单元的隐式代理组装方法：每个单元是基于局部快照训练的LaSDI隐空间常微分方程代理模型，相邻单元通过隐空间中的学习型方向交互项耦合，避免了Schwarz迭代和界面残差评估。采用平滑窗口融合重建全局场，形成可扩展的隐式动力系统。在1维伯格斯与科特韦格-德弗里斯方程上的实验表明，LSEM在训练域外扩展时仍保持预测精度。该方法为构建可解释、可扩展的基于基础模型的代理求解器提供了新路径。
原文摘要 · Abstract (English)
How can we build surrogate solvers that train on small domains but scale to larger ones without intrusive access to PDE operators? Inspired by the Data-Driven Finite Element Method (DD-FEM) framework for modular data-driven solvers, we propose the Latent Space Element Method (LSEM), an element-based latent surrogate assembly approach in which a learned subdomain ("element") model can be tiled and coupled to form a larger computational domain. Each element is a LaSDI latent ODE surrogate trained from snapshots on a local patch, and neighboring elements are coupled through learned directional interaction terms in latent space, avoiding Schwarz iterations and interface residual evaluations. A smooth window-based blending reconstructs a global field from overlapping element predictions, yielding a scalable assembled latent dynamical system. Experiments on the 1D Burgers and Korteweg-de Vries equations show that LSEM maintains predictive accuracy while scaling to spatial domains larger than those seen in training. LSEM offers an interpretable and extensible route toward foundation-model surrogate solvers built from reusable local models.
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