# An operator learning perspective on parameter-to-observable maps

arXiv: https://arxiv.org/abs/2402.06031 · HTML: https://mlsanity.com/paper/2402.06031/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 39 references that are in the map.
- First posted on arXiv 8 Feb 2024.

## Builds on (most cited first, 12 of 12)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Benign Overfitting in Linear Regression](https://mlsanity.com/paper/1906.11300/) (2019)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [A physics-informed operator regression framework for extracting data-driven continuum models](https://mlsanity.com/paper/2009.11992/) (2020)
- [Inverse-design of nonlinear mechanical metamaterials via video denoising diffusion models](https://mlsanity.com/paper/2305.19836/) (2023)
- [Deep learning observables in computational fluid dynamics](https://mlsanity.com/paper/1903.03040/) (2019)
- [Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networks](https://mlsanity.com/paper/2008.05730/) (2020)
- [Elliptic PDE learning is provably data-efficient](https://mlsanity.com/paper/2302.12888/) (2023)
- [Learning High-Dimensional Parametric Maps via Reduced Basis Adaptive Residual Networks](https://mlsanity.com/paper/2112.07096/) (2021)
- [Bayesian inverse problems with unknown operators](https://mlsanity.com/paper/1801.09894/) (2018)

## Built on by (most cited first, 22 of 22)

- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Operator Learning of Lipschitz Operators: An Information-Theoretic Perspective](https://mlsanity.com/paper/2406.18794/) (2024)
- [A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theory](https://mlsanity.com/paper/2510.25379/) (2025)
- [Performance of Neural and Polynomial Operator Surrogates](https://mlsanity.com/paper/2604.00689/) (2026)
- [Learning Memory and Material Dependent Constitutive Laws](https://mlsanity.com/paper/2502.05463/) (2025)
- [Theory-to-Practice Gap for Neural Networks and Neural Operators](https://mlsanity.com/paper/2503.18219/) (2025)
- [One Operator for Many Densities: Amortized Approximation of Conditioning by Neural Operators](https://mlsanity.com/paper/2605.06873/) (2026)
- [Multiple Neural Operators Achieve Near-Optimal Rates for Multi-Task Learning](https://mlsanity.com/paper/2605.22724/) (2026)
- [Generalization Bounds and Statistical Guarantees for Multi-Task and Multiple Operator Learning with MNO Networks](https://mlsanity.com/paper/2604.01961/) (2026)
- [Random feature approximation for general spectral methods](https://mlsanity.com/paper/2506.16283/) (2025)
- [Sequence Modeling with Spectral Mean Flows](https://mlsanity.com/paper/2510.15366/) (2025)
- [Learning WENO for entropy stable schemes to solve conservation laws](https://mlsanity.com/paper/2403.14848/) (2024)
- [Fast Unconstrained Optimization via Hessian Averaging and Adaptive Gradient Sampling Methods](https://mlsanity.com/paper/2408.07268/) (2024)
- [HyperSBINN: A Hypernetwork-Enhanced Systems Biology-Informed Neural Network for Efficient Drug Cardiosafety Assessment](https://mlsanity.com/paper/2408.14266/) (2024)
- [Random Features for Operator-Valued Kernels: Bridging Kernel Methods and Neural Operators](https://mlsanity.com/paper/2603.00971/) (2026)
- [Hamiltonian Neural PDE Solvers through Functional Approximation](https://mlsanity.com/paper/2505.13275/) (2025)
- [Learning Interpretable PDE Representations for Generative Reconstructions with Structured Sparsity](https://mlsanity.com/paper/2604.23867/) (2026)
- [Neural Operators as Efficient Function Interpolators](https://mlsanity.com/paper/2605.07792/) (2026)
- [Universal Approximation of Nonlinear Operators and Their Derivatives](https://mlsanity.com/paper/2605.15285/) (2026)
- [Periodic Neural Mapping for Unsteady Rotor-Blade Pressure and Aeroelastic Load Prediction](https://mlsanity.com/paper/2609.21590/) (2026)
- [Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solver](https://mlsanity.com/paper/2609.24021/) (2026)

## Abstract (arXiv)

Computationally efficient surrogates for parametrized physical models play a crucial role in science and engineering. Operator learning provides data-driven surrogates that map between function spaces. However, instead of full-field measurements, often the available data are only finite-dimensional parametrizations of model inputs or finite observables of model outputs. Building on Fourier Neural Operators, this paper introduces the Fourier Neural Mappings (FNMs) framework that is able to accommodate such finite-dimensional vector inputs or outputs. The paper develops universal approximation theorems for the method. Moreover, in many applications the underlying parameter-to-observable (PtO) map is defined implicitly through an infinite-dimensional operator, such as the solution operator of a partial differential equation. A natural question is whether it is more data-efficient to learn the PtO map end-to-end or first learn the solution operator and subsequently compute the observable from the full-field solution. A theoretical analysis of Bayesian nonparametric regression of linear functionals, which is of independent interest, suggests that the end-to-end approach can actually have worse sample complexity. Extending beyond the theory, numerical results for the FNM approximation of three nonlinear PtO maps demonstrate the benefits of the operator learning perspective that this paper adopts.