mlsanityData to 14 Oct 2026

Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learning

arXiv 2605.22724 ↗First posted 21 May 2026cs.LG cs.NA math.NAcited by 4

Not on the map. Why?

Abstract (arXiv)

We study the approximation and statistical complexity of learning collections of operators in a shared multi-task setting, with a focus on the Multiple Neural Operators (MNO) architecture. For broad classes of Lipschitz multiple operator maps, we derive near-optimal upper bounds for approximation and statistical generalization. On the lower-bound side, we establish a curse of parametric complexity and prove corresponding minimax rates. Together, these results show that shared representations across tasks do not increase the overall cost: multi-task operator learning follows the same scaling laws as single operator learning. We also compare MNO with a multi-task extension of DeepONet based on concatenated task inputs and show that, from a worst-case approximation-complexity perspective, both architectures satisfy essentially the same asymptotic rates.

Key facts. Cited by 4 papers in the map (data through 14 Oct 2026). Builds on 31 of its 55 references that are in the map. First posted on arXiv 21 May 2026.

Builds on

31 of 55 references.

References · 31

  1. 2020Scaling Laws for Neural Language Modelscited by 6,586
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  3. 2020Model Reduction and Neural Networks for Parametric PDEscited by 258
  4. 2024Poseidon: Efficient Foundation Models for PDEscited by 200
  5. 2022MIONet: Learning multiple-input operators via tensor productcited by 181
  6. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 179
  7. 2023In-Context Operator Learning with Data Prompts for Differential Equation Problemscited by 101
  8. 2023Multiple Physics Pretraining for Physical Surrogate Modelscited by 98
  9. 2024Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolationcited by 46
  10. 2022Deep Nonparametric Estimation of Operators between Infinite Dimensional Spacescited by 45
  11. 2021Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equationscited by 44
  12. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
19 more
  1. 2024PROSE-FD: A Multimodal PDE Foundation Model for Learning Multiple Operators for Forecasting Fluid Dynamicscited by 34
  2. 2023Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equationcited by 32
  3. 2023The Parametric Complexity of Operator Learningcited by 28
  4. 2024VICON: Vision In-Context Operator Networks for Multi-Physics Fluid Dynamics Predictioncited by 25
  5. 2023Deep Operator Network Approximation Rates for Lipschitz Operatorscited by 24
  6. 2024An operator learning perspective on parameter-to-observable mapscited by 22
  7. 2022Neural and spectral operator surrogates: unified construction and expression rate boundscited by 21
  8. 2023D2NO: Efficient Handling of Heterogeneous Input Function Spaces with Distributed Deep Neural Operatorscited by 18
  9. 2025BCAT: A Block Causal Transformer for PDE Foundation Models for Fluid Dynamicscited by 18
  10. 2024MODNO: Multi Operator Learning With Distributed Neural Operatorscited by 14
  11. 2024Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Studycited by 14
  12. 2023Globally injective and bijective neural operatorscited by 13
  13. 2024Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learningcited by 12
  14. 2024The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measurescited by 11
  15. 2025A Multimodal PDE Foundation Model for Prediction and Scientific Text Descriptionscited by 10
  16. 2025Probabilistic operator learning: generative modeling and uncertainty quantification for foundation models of differential equationscited by 10
  17. 2024Deep Neural Networks are Adaptive to Function Regularity and Data Distribution in Approximation and Estimationcited by 3
  18. 2025Coefficient-to-Basis Network: A Fine-Tunable Operator Learning Framework for Inverse Problems with Adaptive Discretizations and Theoretical Guaranteescited by 3
  19. 2026Generalization Bounds and Statistical Guarantees for Multi-Task and Multiple Operator Learning with MNO Networkscited by 3

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 4.

4 papers.

  1. 2026Generalization Guarantees for Multi-Input Neural Operator Learning in Sobolev Spacescited by 3
  2. 2026The Information-Theoretic Benefit of Shared Representations under Orthogonality Constraintscited by 0
  3. 2026Kernel Methods for Learning Operators with Multiple Inputs and Outputscited by 0
  4. 2026Function-On-Function Regression Through Separable Neural Operatorscited by 0