Latent Space Element Method
This addresses the challenge of scalable surrogate modeling for PDEs, offering an interpretable and extensible approach for researchers in computational science and engineering, though it appears incremental as it builds on the Data-Driven Finite Element Method framework.
The paper tackles the problem of building surrogate solvers that scale from small to large domains without needing PDE operators, by proposing the Latent Space Element Method (LSEM), which tiles learned element models to form larger computational domains, achieving predictive accuracy on 1D Burgers and Korteweg-de Vries equations while scaling beyond training domains.
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.