# Learning in latent spaces improves the predictive accuracy of deep neural operators

arXiv: https://arxiv.org/abs/2304.07599 · HTML: https://mlsanity.com/paper/2304.07599/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 50 references that are in the map.
- First posted on arXiv 15 Apr 2023.

## Builds on (most cited first, 23 of 23)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [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)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022)
- [Deep transfer operator learning for partial differential equations under conditional shift](https://mlsanity.com/paper/2204.09810/) (2022)
- [Learning two-phase microstructure evolution using neural operators and autoencoder architectures](https://mlsanity.com/paper/2204.07230/) (2022)
- [LNO: Laplace Neural Operator for Solving Differential Equations](https://mlsanity.com/paper/2303.10528/) (2023)
- [Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network](https://mlsanity.com/paper/2201.02217/) (2022)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [On the influence of over-parameterization in manifold based surrogates and deep neural operators](https://mlsanity.com/paper/2203.05071/) (2022)
- [Bi-fidelity Modeling of Uncertain and Partially Unknown Systems using DeepONets](https://mlsanity.com/paper/2204.00997/) (2022)
- [A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems](https://mlsanity.com/paper/2202.04648/) (2022)
- [Extending classical surrogate modelling to high-dimensions through supervised dimensionality reduction: a data-driven approach](https://mlsanity.com/paper/1812.06309/) (2018)
- [Manifold learning-based polynomial chaos expansions for high-dimensional surrogate models](https://mlsanity.com/paper/2107.09814/) (2021)
- [Neural operator learning of heterogeneous mechanobiological insults contributing to aortic aneurysms](https://mlsanity.com/paper/2205.03780/) (2022)
- [MultiAuto-DeepONet: A Multi-resolution Autoencoder DeepONet for Nonlinear Dimension Reduction, Uncertainty Quantification and Operator Learning of Forward and Inverse Stochastic Problems](https://mlsanity.com/paper/2204.03193/) (2022)

## Built on by (most cited first, 19 of 19)

- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [Generative Latent Neural PDE Solver using Flow Matching](https://mlsanity.com/paper/2503.22600/) (2025)
- [A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domains](https://mlsanity.com/paper/2405.12465/) (2024)
- [Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Study](https://mlsanity.com/paper/2410.00357/) (2024)
- [Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learning](https://mlsanity.com/paper/2401.10490/) (2024)
- [A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformations](https://mlsanity.com/paper/2404.00074/) (2024)
- [Solving multiphysics-based inverse problems with learned surrogates and constraints](https://mlsanity.com/paper/2307.11099/) (2023)
- [Size Lowerbounds for Deep Operator Networks](https://mlsanity.com/paper/2308.06338/) (2023)
- [An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learning](https://mlsanity.com/paper/2408.13222/) (2024)
- [LatentPINNs: Generative physics-informed neural networks via a latent representation learning](https://mlsanity.com/paper/2305.07671/) (2023)
- [FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulation](https://mlsanity.com/paper/2505.20349/) (2025)
- [Integration of physics-informed operator learning and finite element method for parametric learning of partial differential equations](https://mlsanity.com/paper/2401.02363/) (2024)
- [Evaluation of Deep Neural Operator Models toward Ocean Forecasting](https://mlsanity.com/paper/2308.11814/) (2023)
- [Introducing a microstructure-embedded autoencoder approach for reconstructing high-resolution solution field data from a reduced parametric space](https://mlsanity.com/paper/2405.01975/) (2024)
- [Coefficient-to-Basis Network: A Fine-Tunable Operator Learning Framework for Inverse Problems with Adaptive Discretizations and Theoretical Guarantees](https://mlsanity.com/paper/2503.08642/) (2025)
- [A Neural Operator Emulator for Coastal and Riverine Shallow Water Dynamics](https://mlsanity.com/paper/2502.14782/) (2025)
- [Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators](https://mlsanity.com/paper/2509.26371/) (2025)
- [Deep Adaptive Dimension Reduction for Bayesian Inference in Inverse Problems](https://mlsanity.com/paper/2605.29373/) (2026)
- [Spectrally Regularized Latent Flow Matching for Turbulence Generation](https://mlsanity.com/paper/2606.11691/) (2026)

## Abstract (arXiv)

Operator regression provides a powerful means of constructing discretization-invariant emulators for partial-differential equations (PDEs) describing physical systems. Neural operators specifically employ deep neural networks to approximate mappings between infinite-dimensional Banach spaces. As data-driven models, neural operators require the generation of labeled observations, which in cases of complex high-fidelity models result in high-dimensional datasets containing redundant and noisy features, which can hinder gradient-based optimization. Mapping these high-dimensional datasets to a low-dimensional latent space of salient features can make it easier to work with the data and also enhance learning. In this work, we investigate the latent deep operator network (L-DeepONet), an extension of standard DeepONet, which leverages latent representations of high-dimensional PDE input and output functions identified with suitable autoencoders. We illustrate that L-DeepONet outperforms the standard approach in terms of both accuracy and computational efficiency across diverse time-dependent PDEs, e.g., modeling the growth of fracture in brittle materials, convective fluid flows, and large-scale atmospheric flows exhibiting multiscale dynamical features.