# Variational Autoencoding Neural Operators

arXiv: https://arxiv.org/abs/2302.10351 · HTML: https://mlsanity.com/paper/2302.10351/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 63 references that are in the map.
- First posted on arXiv 20 Feb 2023.

## Builds on (most cited first, 30 of 37)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Instant Neural Graphics Primitives with a Multiresolution Hash Encoding](https://mlsanity.com/paper/2201.05989/) (2022)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Understanding disentangling in β-VAE](https://mlsanity.com/paper/1804.03599/) (2018)
- [Neural Processes](https://mlsanity.com/paper/1807.01622/) (2018)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Attentive Neural Processes](https://mlsanity.com/paper/1901.05761/) (2019)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Adversarial Generation of Continuous Images](https://mlsanity.com/paper/2011.12026/) (2020)
- [NeRF-VAE: A Geometry Aware 3D Scene Generative Model](https://mlsanity.com/paper/2104.00587/) (2021)
- [Image Generators with Conditionally-Independent Pixel Synthesis](https://mlsanity.com/paper/2011.13775/) (2020)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [Generative Models as Distributions of Functions](https://mlsanity.com/paper/2102.04776/) (2021)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Generative Adversarial Neural Operators](https://mlsanity.com/paper/2205.03017/) (2022)

## Built on by (most cited first, 22 of 22)

- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [CViT: Continuous Vision Transformer for Operator Learning](https://mlsanity.com/paper/2405.13998/) (2024)
- [Derivative-informed neural operator acceleration of geometric MCMC for infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2403.08220/) (2024)
- [Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning](https://mlsanity.com/paper/2506.10973/) (2025)
- [Mesh-Informed Neural Operator : A Transformer Generative Approach](https://mlsanity.com/paper/2506.16656/) (2025)
- [Universal Functional Regression with Neural Operator Flows](https://mlsanity.com/paper/2404.02986/) (2024)
- [FUSE: Fast Unified Simulation and Estimation for PDEs](https://mlsanity.com/paper/2405.14558/) (2024)
- [Generative Adversarial Reduced Order Modelling](https://mlsanity.com/paper/2305.15881/) (2023)
- [LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transport](https://mlsanity.com/paper/2411.12726/) (2024)
- [LatentPINNs: Generative physics-informed neural networks via a latent representation learning](https://mlsanity.com/paper/2305.07671/) (2023)
- [FNOPE: Simulation-based inference on function spaces with Fourier Neural Operators](https://mlsanity.com/paper/2505.22573/) (2025)
- [Shape-Informed Clustering of Multi-Dimensional Functional Data via Deep Functional Autoencoders](https://mlsanity.com/paper/2509.22969/) (2025)
- [Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling](https://mlsanity.com/paper/2510.00233/) (2025)
- [Deep set based operator learning with uncertainty quantification](https://mlsanity.com/paper/2509.25646/) (2025)
- [Multimodal Scientific Learning Beyond Diffusions and Flows](https://mlsanity.com/paper/2602.00960/) (2026)
- [Hypernetwork-based approach for grid-independent functional data clustering](https://mlsanity.com/paper/2602.22823/) (2026)
- [SA-EMO: Structure-Aligned Encoder Mixture of Operators for Generalizable Full-waveform Inversion](https://mlsanity.com/paper/2511.11627/) (2025)
- [GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Media](https://mlsanity.com/paper/2602.14642/) (2026)
- [SPAMoE: Spectrum-Aware Hybrid Operator Framework for Full-Waveform Inversion](https://mlsanity.com/paper/2604.07421/) (2026)
- [NOFE - Neural Operator Function Embedding](https://mlsanity.com/paper/2605.11970/) (2026)
- [No Free Lunch in Flow Surrogates under Time-Varying Boundary Conditions: A Two-Regime Study](https://mlsanity.com/paper/2607.23667/) (2026)

## Abstract (arXiv)

Unsupervised learning with functional data is an emerging paradigm of machine learning research with applications to computer vision, climate modeling and physical systems. A natural way of modeling functional data is by learning operators between infinite dimensional spaces, leading to discretization invariant representations that scale independently of the sample grid resolution. Here we present Variational Autoencoding Neural Operators (VANO), a general strategy for making a large class of operator learning architectures act as variational autoencoders. For this purpose, we provide a novel rigorous mathematical formulation of the variational objective in function spaces for training. VANO first maps an input function to a distribution over a latent space using a parametric encoder and then decodes a sample from the latent distribution to reconstruct the input, as in classic variational autoencoders. We test VANO with different model set-ups and architecture choices for a variety of benchmarks. We start from a simple Gaussian random field where we can analytically track what the model learns and progressively transition to more challenging benchmarks including modeling phase separation in Cahn-Hilliard systems and real world satellite data for measuring Earth surface deformation.