# Generalization Bounds and Statistical Guarantees for Multi-Task and Multiple Operator Learning with MNO Networks

arXiv: https://arxiv.org/abs/2604.01961 · HTML: https://mlsanity.com/paper/2604.01961/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 71 references that are in the map.
- First posted on arXiv 2 Apr 2026.

## Builds on (most cited first, 30 of 43)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [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)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Speech Recognition with Deep Recurrent Neural Networks](https://mlsanity.com/paper/1303.5778/) (2013)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [Multiple Physics Pretraining for Physical Surrogate Models](https://mlsanity.com/paper/2310.02994/) (2023)
- [Nonparametric Regression on Low-Dimensional Manifolds using Deep ReLU Networks : Function Approximation and Statistical Recovery](https://mlsanity.com/paper/1908.01842/) (2019)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness](https://mlsanity.com/paper/2305.17289/) (2023)
- [Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration](https://mlsanity.com/paper/2303.04778/) (2023)
- [Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation](https://mlsanity.com/paper/2404.12355/) (2024)
- [Deep Nonparametric Estimation of Operators between Infinite Dimensional Spaces](https://mlsanity.com/paper/2201.00217/) (2022)
- [Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equations](https://mlsanity.com/paper/2112.08125/) (2021)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [PROSE-FD: A Multimodal PDE Foundation Model for Learning Multiple Operators for Forecasting Fluid Dynamics](https://mlsanity.com/paper/2409.09811/) (2024)
- [Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation](https://mlsanity.com/paper/2301.11509/) (2023)
- [The Parametric Complexity of Operator Learning](https://mlsanity.com/paper/2306.15924/) (2023)
- [Learning Green's functions associated with time-dependent partial differential equations](https://mlsanity.com/paper/2204.12789/) (2022)
- [VICON: Vision In-Context Operator Networks for Multi-Physics Fluid Dynamics Prediction](https://mlsanity.com/paper/2411.16063/) (2024)
- [Deep Operator Network Approximation Rates for Lipschitz Operators](https://mlsanity.com/paper/2307.09835/) (2023)
- [An operator learning perspective on parameter-to-observable maps](https://mlsanity.com/paper/2402.06031/) (2024)
- [Neural and spectral operator surrogates: unified construction and expression rate bounds](https://mlsanity.com/paper/2207.04950/) (2022)

## Built on by (most cited first, 3 of 3)

- [Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learning](https://mlsanity.com/paper/2605.22724/) (2026)
- [Generalization Guarantees for Multi-Input Neural Operator Learning in Sobolev Spaces](https://mlsanity.com/paper/2606.17419/) (2026)
- [Kernel Methods for Learning Operators with Multiple Inputs and Outputs](https://mlsanity.com/paper/2608.11831/) (2026)

## Abstract (arXiv)

Multiple operator learning concerns learning operator families G[α]:U→ Vα∈ W indexed by an operator descriptor α. Training data are collected hierarchically by sampling operator instances α, then input functions u per instance, and finally evaluation points x per input, yielding noisy observations of G[α][u](x). While recent work has developed expressive multi-task and multiple operator learning architectures and approximation-theoretic scaling laws, quantitative statistical generalization guarantees remain limited. We provide a covering-number-based generalization analysis for separable models, focusing on the Multiple Neural Operator (MNO) architecture: we first derive explicit metric-entropy bounds for hypothesis classes given by linear combinations of products of deep ReLU subnetworks, and then combine these complexity bounds with approximation guarantees for MNO to obtain an explicit approximation-estimation tradeoff for the expected test error on new (unseen) triples (α,u,x). The resulting bound makes the dependence on the hierarchical sampling budgets (nα,nu,nx) transparent and yields an explicit learning-rate statement in the operator-sampling budget nα, providing a sample-complexity characterization for generalization across operator instances. The structure and architecture can also be viewed as a general purpose solver or an example of a "small'' PDE foundation model, where the triples are one form of multi-modality.