Latent Space Element Method

2026-01-05

AI Summary

This study addresses the challenge of constructing surrogate solvers that can be trained on small subdomains and seamlessly extended to large spatial domains without requiring access to the underlying partial differential equation (PDE) operators. The authors propose a novel approach based on assembling local latent-space elements: LaSDI latent-space ODE models are trained on subdomains and coupled through learned directional interaction terms between neighboring elements, while a windowed weighting strategy ensures smooth global field reconstruction. This work introduces, for the first time, a finite element–inspired assembly paradigm into latent-space modeling, thereby eliminating the need for iterative solvers or interface residual computations. The resulting framework is modular, reusable, and interpretable. Experiments on the 1D Burgers and Korteweg–de Vries equations demonstrate that the model maintains high accuracy and strong scalability even when extrapolating to spatial domains significantly larger than those seen during training.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• surrogate solver
• domain scaling
• non-intrusive PDE
• latent space
• modular modeling

Innovation

• Latent Space Element Method
• surrogate solver
• domain decomposition
• latent ODE
• scalable PDE emulation