# Latent Space Element Method

arXiv: https://arxiv.org/abs/2601.01741 · HTML: https://mlsanity.com/paper/2601.01741/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 24 references that are in the map.
- First posted on arXiv 5 Jan 2026.

## Builds on (most cited first, 13 of 13)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Mosaic Flows: A Transferable Deep Learning Framework for Solving PDEs on Unseen Domains](https://mlsanity.com/paper/2104.10873/) (2021)
- [GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder](https://mlsanity.com/paper/2308.05882/) (2023)
- [tLaSDI: Thermodynamics-informed latent space dynamics identification](https://mlsanity.com/paper/2403.05848/) (2024)
- [A Comprehensive Review of Latent Space Dynamics Identification Algorithms for Intrusive and Non-Intrusive Reduced-Order-Modeling](https://mlsanity.com/paper/2403.10748/) (2024)
- [Weak-Form Latent Space Dynamics Identification](https://mlsanity.com/paper/2311.12880/) (2023)
- [Defining Foundation Models for Computational Science: A Call for Clarity and Rigor](https://mlsanity.com/paper/2505.22904/) (2025)
- [Component-wise reduced order model lattice-type structure design](https://mlsanity.com/paper/2010.10770/) (2020)
- [Train Small, Model Big: Scalable Physics Simulators via Reduced Order Modeling and Domain Decomposition](https://mlsanity.com/paper/2401.10245/) (2023)
- [Neural-operator element method: Efficient and scalable finite element method enabled by reusable neural operators](https://mlsanity.com/paper/2506.18427/) (2025)
- [The role of interface boundary conditions and sampling strategies for Schwarz-based coupling of projection-based reduced order models](https://mlsanity.com/paper/2410.04668/) (2024)
- [Sequential decoder training for improved latent space dynamics identification](https://mlsanity.com/paper/2510.03535/) (2025)
- [Higher-Order LaSDI: Reduced Order Modeling with Multiple Time Derivatives](https://mlsanity.com/paper/2512.15997/) (2025)

## Built on by (most cited first, 1 of 1)

- [Convex Neural Energy Elements: Monolithic Finite-Element Assembly of Geometry-Parameterized Neural Operators with Stability and Error Guarantees](https://mlsanity.com/paper/2608.02036/) (2026)

## Abstract (arXiv)

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.