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

arXiv: https://arxiv.org/abs/2411.18014 · HTML: https://mlsanity.com/paper/2411.18014/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 14 references that are in the map.
- First posted on arXiv 27 Nov 2024.

## Builds on (most cited first, 7 of 7)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Neural Operators with Localized Integral and Differential Kernels](https://mlsanity.com/paper/2402.16845/) (2024)
- [DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domains](https://mlsanity.com/paper/2402.07250/) (2024)
- [Graph Fourier Neural Kernels (G-FuNK): Learning Solutions of Nonlinear Diffusive Parametric PDEs on Multiple Domains](https://mlsanity.com/paper/2410.04655/) (2024)

## Built on by (most cited first, 5 of 5)

- [Geometric Operator Learning with Optimal Transport](https://mlsanity.com/paper/2507.20065/) (2025)
- [FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learning](https://mlsanity.com/paper/2509.12344/) (2025)
- [A unified framework for geometry-independent operator learning in cardiac electrophysiology simulations](https://mlsanity.com/paper/2512.01702/) (2025)
- [Geometry-physics confounding impairs PDE learning across varying domains](https://mlsanity.com/paper/2609.38623/) (2026)
- [Advectra: Asymmetric Latent Transport for Non-Stationary Physics](https://mlsanity.com/paper/2610.05098/) (2026)

## Abstract (arXiv)

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