# Deep Gaussian Processes for Functional Maps

arXiv: https://arxiv.org/abs/2510.22068 · HTML: https://mlsanity.com/paper/2510.22068/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 59 references that are in the map.
- First posted on arXiv 24 Oct 2025.

## Builds on (most cited first, 25 of 25)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [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)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Doubly Stochastic Variational Inference for Deep Gaussian Processes](https://mlsanity.com/paper/1705.08933/) (2017)
- [Functional Regression](https://mlsanity.com/paper/1406.4068/) (2014)
- [Multi-Resolution Active Learning of Fourier Neural Operators](https://mlsanity.com/paper/2309.16971/) (2023)
- [Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains](https://mlsanity.com/paper/2305.19663/) (2023)
- [AutoIP: A United Framework to Integrate Physics into Gaussian Processes](https://mlsanity.com/paper/2202.12316/) (2022)
- [Functional Linear Regression with Mixed Predictors](https://mlsanity.com/paper/2012.00460/) (2020)
- [Additive Function-on-Function Regression](https://mlsanity.com/paper/1606.03775/) (2016)
- [Robust Function-on-Function Regression](https://mlsanity.com/paper/1908.11601/) (2019)
- [On function-on-function regression: Partial least squares approach](https://mlsanity.com/paper/1912.06995/) (2019)
- [Variable selection in functional regression models: a review](https://mlsanity.com/paper/2401.14867/) (2024)

## Abstract (arXiv)

Learning mappings between functional spaces, also known as function-on-function regression, is a fundamental problem in functional data analysis with broad applications, including spatiotemporal forecasting, curve prediction, and climate modeling. Existing approaches often struggle to capture complex nonlinear relationships and/or provide reliable uncertainty quantification when data are noisy, sparse, or irregularly sampled. To address these challenges, we propose Deep Gaussian Processes for Functional Maps (DGPFM). Our method constructs a sequence of GP-based linear and nonlinear transformations directly in function space, leveraging kernel integral transforms, GP conditional means, and nonlinear activations sampled from Gaussian processes. A key insight enables a simplified and flexible implementation: under fixed evaluation locations, discrete approximations of kernel integral transforms reduce to direct functional integral transforms, allowing seamless integration of diverse transform designs. To support scalable probabilistic inference, we adopt inducing points and whitening transformations within a variational learning framework. Empirical results on both real-world and synthetic benchmark datasets demonstrate the advantages of DGPFM in terms of predictive accuracy and uncertainty calibration.