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.

翻译：如何构建在小型域上训练但无需侵入式访问PDE算子即可扩展至更大域的代理求解器？受模块化数据驱动求解器的数据驱动有限元法（DD-FEM）框架启发，我们提出潜在空间单元法（LSEM）——一种基于单元的潜在代理组装方法，其中习得的子域（“单元”）模型可通过平铺与耦合形成更大的计算域。每个单元均为在局部区块上通过快照训练的LaSDI潜在常微分方程代理，相邻单元通过潜在空间中习得的方向性交互项进行耦合，从而避免了施瓦茨迭代与界面残差评估。基于平滑窗口的融合方法从重叠的单元预测中重构全局场，形成可扩展的组装潜在动力系统。在一维Burgers方程和Korteweg-de Vries方程上的实验表明，LSEM在扩展至训练未见更大空间域时仍保持预测精度。LSEM为构建基于可复用局部模型的基础模型代理求解器提供了一条可解释且可扩展的路径。