Latent Space Element Method

TL;DR

We address scalable PDE surrogate modeling with non-intrusive, geometry-flexible surrogates by introducing Latent Space Element Method (LSEM), which assembles LaSDI latent-ODE element models through smooth window blending and learned inter-element interactions. Each element carries a reusable latent-space dynamic learned from local patches, and the global evolution is formed by a block-structured latent operator that couples neighboring elements, avoiding intrusive solvers. Stability and robustness are enforced via a reparameterization of latent states and regularization on the latent dynamics, enabling accurate prediction on domains larger than training data while maintaining efficiency. Empirical results on 1D Burgers' and KdV equations demonstrate strong geometric scalability and substantial speedups (tens to thousands of times faster than full-order models) with competitive accuracy, validating the potential of foundation-model–style surrogates for computational science. The work highlights a practical blueprint for non-intrusive, reusable local models that scale to complex geometries and larger domains, supporting future extensions to higher dimensions and uncertainty quantification.

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.

Figures (10)

• Figure 1: Schematic diagram of latent dynamics identification framework: The autoencoder maps the high-dimensional state $\mathbf{q}$ to a low-dimensional latent variable $\mathbf{z}$, whose evolution is governed by the learned latent ODE $\dot{\mathbf{z}} = \bm{\Xi}\bm{\Phi}(\mathbf{z})$. The predicted latent trajectory is then decoded to reconstruct the physical state, enabling efficient reduced-order modeling of complex dynamical systems.
• Figure 2: Schematic diagram of interface handling in the proposed LSEM framework: Local autoencoders are assigned to reference subdomain types, and global configurations are constructed by tiling these subdomains with overlapping regions. Smooth window functions $W(\mathbf{x})$ blend the reconstructed subdomain solutions, ensuring continuity and smooth transitions across element interfaces.
• Figure 3: Schematic diagram of interaction dynamics modeling between neighboring latent-space elements: Each element’s latent variable evolves according to its internal dynamics together with contributions from neighboring elements, represented by directional coefficient blocks (e.g., $\bm{\Xi}_t$, $\bm{\Xi}_l$, $\bm{\Xi}_{br}$). This structure enables the model to capture information flow across elements and assemble a coherent global latent dynamics operator.
• Figure 4: Four-element training datasets for one-dimensional Burgers' equation. White lines indicate element interfaces.
• Figure 5: Solution composition on the four-element domain: (a) window functions $W_i(x)$ at each element $i=1,\ldots,4$; and (b) element-wise contribution $W_i(x)u_i(x)$, with black dashed line showing the global solution $u(x) = \sum_{i}W_i(x)u_i(x)$. Gray lines mark the element interfaces.