Diffeomorphic Latent Neural Operators for Data-Efficient Learning of Solutions to Partial Differential Equations

TL;DR

This work proposes that in order to learn a PDE solution operator that can generalize across multiple domains without needing to sample enough data expressive enough for all possible geometries, a latent neural operator can be trained on just a few ground truth solution fields diffeomorphically mapped from different geometric/spatial domains to a fixed reference configuration.

Abstract

A computed approximation of the solution operator to a system of partial differential equations (PDEs) is needed in various areas of science and engineering. Neural operators have been shown to be quite effective at predicting these solution generators after training on high-fidelity ground truth data (e.g. numerical simulations). However, in order to generalize well to unseen spatial domains, neural operators must be trained on an extensive amount of geometrically varying data samples that may not be feasible to acquire or simulate in certain contexts (e.g., patient-specific medical data, large-scale computationally intensive simulations.) We propose that in order to learn a PDE solution operator that can generalize across multiple domains without needing to sample enough data expressive enough for all possible geometries, we can train instead a latent neural operator on just a few ground truth solution fields diffeomorphically mapped from different geometric/spatial domains to a fixed reference configuration. Furthermore, the form of the solutions is dependent on the choice of mapping to and from the reference domain. We emphasize that preserving properties of the differential operator when constructing these mappings can significantly reduce the data requirement for achieving an accurate model due to the regularity of the solution fields that the latent neural operator is training on. We provide motivating numerical experimentation that demonstrates an extreme case of this consideration by exploiting the conformal invariance of the Laplacian

Figures (4)

• Figure 1: A schematic outlining the construction of the latent operator $\mathcal{F}_0$ within the diffeomorphic mapping operator learning framework.
• Figure 2: Schematics for $\varphi^{-1}$ computation via the extermap function in sc-toolbox.
• Figure 3: Comparison of mapping approaches (SC-Map, LDDMM, and Discrete OT) for the Laplace solutions on doubly connected domains. The first three columns illustrate the ground truth solution, predicted solution, and the error (ground truth - prediction) for each mapping approach. The rightmost panel shows the relative $L_2$ error as a function of the number of training samples, highlighting the superior performance and efficiency of SC-Map compared to LDDMM.
• Figure 4: Visual comparison of mappings $\varphi_\alpha : \Omega_\alpha \to \Omega_0$ for different approaches—LDDMM, Schwarz-Christoffel (SC) map, and discrete optimal transport (OT)—and their impact on the Laplace solution. The true Laplace solution on the original domain $\Omega_\alpha$ is mapped to the reference domain $\Omega_0$ using each mapping method. The SC map preserves the Laplace operator, resulting in an exact solution on $\Omega_0$, while LDDMM introduces minor distortions, and discrete OT introduces significant artifacts due to its non-diffeomorphic nature. The rightmost solution represents the true Laplace solution on the annulus $\Omega_0$, illustrating the fidelity of each mapping.