# The Parametric Complexity of Operator Learning

arXiv: https://arxiv.org/abs/2306.15924 · HTML: https://mlsanity.com/paper/2306.15924/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jun 2023.

## Built on by (most cited first, 28 of 28)

- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Deep Operator Network Approximation Rates for Lipschitz Operators](https://mlsanity.com/paper/2307.09835/) (2023)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Mixtures of Neural Operators Reduce Active Complexity in Operator Learning](https://mlsanity.com/paper/2404.09101/) (2024)
- [Neural Scaling Laws of Deep ReLU and Deep Operator Network: A Theoretical Study](https://mlsanity.com/paper/2410.00357/) (2024)
- [The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measures](https://mlsanity.com/paper/2410.23440/) (2024)
- [A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theory](https://mlsanity.com/paper/2510.25379/) (2025)
- [Towards Sharp Minimax Risk Bounds for Operator Learning](https://mlsanity.com/paper/2512.17805/) (2025)
- [One model to solve them all: 2BSDE families via neural operators](https://mlsanity.com/paper/2511.01125/) (2025)
- [Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learning](https://mlsanity.com/paper/2605.22724/) (2026)
- [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)
- [Deep Neural Operator Learning for Probabilistic Models](https://mlsanity.com/paper/2511.07235/) (2025)
- [Generalization Bounds and Statistical Guarantees for Multi-Task and Multiple Operator Learning with MNO Networks](https://mlsanity.com/paper/2604.01961/) (2026)
- [Polynomial Scaling is Possible For Neural Operator Approximations of Structured Families of BSDEs](https://mlsanity.com/paper/2410.14788/) (2024)
- [Operator Learning for Families of Finite-State Mean-Field Games](https://mlsanity.com/paper/2602.13169/) (2026)
- [Upper Approximation Bounds for Neural Oscillators](https://mlsanity.com/paper/2512.01015/) (2025)
- [Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEs](https://mlsanity.com/paper/2601.06388/) (2026)
- [Adaptivity Under Realizability Constraints: Comparing In-Context and Agentic Learning](https://mlsanity.com/paper/2605.04995/) (2026)
- [Variation Spaces for Encoder--Decoder Neural Operators: Approximation and Generalization](https://mlsanity.com/paper/2606.01244/) (2026)
- [Predictor-Based Output-Feedback Control of Linear Systems with Time-Varying Input and Measurement Delays via Neural-Approximated Prediction Horizons](https://mlsanity.com/paper/2603.29117/) (2026)
- [Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation](https://mlsanity.com/paper/2605.08170/) (2026)
- [Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs](https://mlsanity.com/paper/2605.10277/) (2026)
- [HAMNO: A Hierarchical Adaptive Multi-scale Neural Operator with Physics-Informed Learning for Dynamical Systems](https://mlsanity.com/paper/2606.11963/) (2026)
- [Fourier Neural Operators with Least-Squares Readout Refit for Learning Random Obstacle-to-Solution Maps](https://mlsanity.com/paper/2606.29436/) (2026)
- [NeuralChaos: Optimal Adapted Approximation of Square Integrable Predictable Processes](https://mlsanity.com/paper/2607.14361/) (2026)
- [Neural operators approximate strongly continuous convex monotone semigroups](https://mlsanity.com/paper/2609.02727/) (2026)
- [Neural Scaling Laws of Transformer Operator Network](https://mlsanity.com/paper/2609.33533/) (2026)
- [Into the danger zone: stable extrapolation in high-dimensional function and operator learning](https://mlsanity.com/paper/2609.36709/) (2026)

## Abstract (arXiv)

Neural operator architectures employ neural networks to approximate operators mapping between Banach spaces of functions; they may be used to accelerate model evaluations via emulation, or to discover models from data. Consequently, the methodology has received increasing attention over recent years, giving rise to the rapidly growing field of operator learning. The first contribution of this paper is to prove that for general classes of operators which are characterized only by their Cr- or Lipschitz-regularity, operator learning suffers from a "curse of parametric complexity", which is an infinite-dimensional analogue of the well-known curse of dimensionality encountered in high-dimensional approximation problems. The result is applicable to a wide variety of existing neural operators, including PCA-Net, DeepONet and the FNO.The second contribution of the paper is to prove that this general curse can be overcome for solution operators defined by the Hamilton-Jacobi equation; this is achieved by leveraging additional structure in the underlying solution operator, going beyond regularity. To this end, a novel neural operator architecture is introduced, termed HJ-Net, which explicitly takes into account characteristic information of the underlying Hamiltonian system. Error and complexity estimates are derived for HJ-Net which show that this architecture can provably beat the curse of parametric complexity related to the infinite-dimensional input and output function spaces.