# Learning Neural Operators from Partial Observations via Latent Autoregressive Modeling

arXiv: https://arxiv.org/abs/2601.15547 · HTML: https://mlsanity.com/paper/2601.15547/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 51 references that are in the map.
- First posted on arXiv 22 Jan 2026.

## Builds on (most cited first, 30 of 35)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [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)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [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)
- [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)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [A disciplined approach to neural network hyper-parameters: Part 1 -- learning rate, batch size, momentum, and weight decay](https://mlsanity.com/paper/1803.09820/) (2018)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [Operator Learning with Neural Fields: Tackling PDEs on General Geometries](https://mlsanity.com/paper/2306.07266/) (2023)
- [Latent Neural Operator for Solving Forward and Inverse PDE Problems](https://mlsanity.com/paper/2406.03923/) (2024)
- [Improved Operator Learning by Orthogonal Attention](https://mlsanity.com/paper/2310.12487/) (2023)
- [Task Transformer Network for Joint MRI Reconstruction and Super-Resolution](https://mlsanity.com/paper/2106.06742/) (2021)
- [Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning](https://mlsanity.com/paper/2402.15734/) (2024)
- [VFlow: Discovering Optimal Agentic Workflows for Verilog Generation](https://mlsanity.com/paper/2504.03723/) (2025)
- [Inducing Point Operator Transformer: A Flexible and Scalable Architecture for Solving PDEs](https://mlsanity.com/paper/2312.10975/) (2023)
- [Accelerating PDE Data Generation via Differential Operator Action in Solution Space](https://mlsanity.com/paper/2402.05957/) (2024)
- [Mixture-of-Experts Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2510.25803/) (2025)
- [Deep learning electromagnetic inversion with convolutional neural networks](https://mlsanity.com/paper/1812.10247/) (2018)

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

- [Spatial Context or Memory? Sizing the Inputs of Neural Surrogates for Partially Observed Fields](https://mlsanity.com/paper/2610.12723/) (2026)

## Abstract (arXiv)

Real-world scientific applications frequently encounter incomplete observational data due to sensor limitations, geographic constraints, or measurement costs. Although neural operators significantly advanced PDE solving in terms of computational efficiency and accuracy, their underlying assumption of fully-observed spatial inputs severely restricts applicability in real-world applications. We introduce the first systematic framework for learning neural operators from partial observation. We identify and formalize two fundamental obstacles: (i) the supervision gap in unobserved regions that prevents effective learning of physical correlations, and (ii) the dynamic spatial mismatch between incomplete inputs and complete solution fields. Specifically, our proposed Latent Autoregressive Neural Operator(LANO) introduces two novel components designed explicitly to address the core difficulties of partial observations: (i) a mask-to-predict training strategy that creates artificial supervision by strategically masking observed regions, and (ii) a Physics-Aware Latent Propagator that reconstructs solutions through boundary-first autoregressive generation in latent space. Additionally, we develop POBench-PDE, a dedicated and comprehensive benchmark designed specifically for evaluating neural operators under partial observation conditions across three PDE-governed tasks. LANO achieves state-of-the-art performance with 18--69% relative L2 error reduction across all benchmarks under patch-wise missingness with less than 50% missing rate, including real-world climate prediction. Our approach effectively addresses practical scenarios involving up to 75% missing rate, to some extent bridging the existing gap between idealized research settings and the complexities of real-world scientific computing.