Latent Space Element Method

TL;DR

The paper introduces the Latent Space Element Method (LSEM), a scalable surrogate modeling approach that assembles local learned models to predict solutions on larger domains without intrusive PDE access.

Contribution

LSEM presents a novel element-based latent surrogate assembly framework that enables scalable, non-intrusive PDE solution approximation using learned local models coupled in latent space.

Findings

LSEM maintains accuracy on larger domains than training data.

LSEM avoids Schwarz iterations and interface residual evaluations.

LSEM demonstrates effectiveness on 1D Burgers and KdV equations.

Abstract

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.