# Rates and architectures for learning geometrically non-trivial operators

arXiv: https://arxiv.org/abs/2512.09376 · HTML: https://mlsanity.com/paper/2512.09376/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 72 references that are in the map.
- First posted on arXiv 10 Dec 2025.

## Builds on (most cited first, 30 of 38)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Nerfies: Deformable Neural Radiance Fields](https://mlsanity.com/paper/2011.12948/) (2020)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Implicit Geometric Regularization for Learning Shapes](https://mlsanity.com/paper/2002.10099/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [WIRE: Wavelet Implicit Neural Representations](https://mlsanity.com/paper/2301.05187/) (2023)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Deep Level Sets: Implicit Surface Representations for 3D Shape Inference](https://mlsanity.com/paper/1901.06802/) (2019)
- [Data-driven discovery of Green's functions with human-understandable deep learning](https://mlsanity.com/paper/2105.00266/) (2021)
- [Learning elliptic partial differential equations with randomized linear algebra](https://mlsanity.com/paper/2102.00491/) (2021)
- [Learning Green's functions associated with time-dependent partial differential equations](https://mlsanity.com/paper/2204.12789/) (2022)
- [Elliptic PDE learning is provably data-efficient](https://mlsanity.com/paper/2302.12888/) (2023)
- [Bridging Physics-based and Data-driven modeling for Learning Dynamical Systems](https://mlsanity.com/paper/2011.10616/) (2020)
- [Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learning](https://mlsanity.com/paper/2201.12354/) (2022)
- [Neural Integral Equations](https://mlsanity.com/paper/2209.15190/) (2022)
- [Deep Neural Networks Are Effective At Learning High-Dimensional Hilbert-Valued Functions From Limited Data](https://mlsanity.com/paper/2012.06081/) (2020)
- [Solving Electrical Impedance Tomography with Deep Learning](https://mlsanity.com/paper/1906.03944/) (2019)
- [Deep Operator Learning Lessens the Curse of Dimensionality for PDEs](https://mlsanity.com/paper/2301.12227/) (2023)

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

- [Optimal Multiscale Learning of Linear Operators](https://mlsanity.com/paper/2606.16913/) (2026)

## Abstract (arXiv)

Deep learning methods have proven capable of recovering operators between high-dimensional spaces, such as solution maps of PDEs and similar objects in mathematical physics, from very few training samples. This phenomenon of data-efficiency has been proven for certain classes of elliptic operators with simple geometry, i.e., operators that do not change the domain of the function or propagate singularities. However, scientific machine learning is commonly used for problems that do involve the propagation of singularities in a priori unknown ways, such as waves, advection, and fluid dynamics. In light of this, we expand the learning theory to include double fibration transforms--geometric integral operators that include generalized Radon and geodesic ray transforms. We prove that this class of operators does not suffer from the curse of dimensionality: the error decays superalgebraically, that is, faster than any fixed power of the reciprocal of the number of training samples. Furthermore, we investigate architectures that explicitly encode the geometry of these transforms, demonstrating that an architecture reminiscent of cross-attention based on levelset methods yields a parameterization that is universal, stable, and learns double fibration transforms from very few training examples. Our results contribute to a rapidly-growing line of theoretical work on learning operators for scientific machine learning.