# Nonparametric learning of kernels in nonlocal operators

arXiv: https://arxiv.org/abs/2205.11006 · HTML: https://mlsanity.com/paper/2205.11006/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 76 references that are in the map.
- First posted on arXiv 23 May 2022.

## Builds on (most cited first, 24 of 24)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Non-local Neural Networks](https://mlsanity.com/paper/1711.07971/) (2017)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Neural Networks as Kernel Learners: The Silent Alignment Effect](https://mlsanity.com/paper/2111.00034/) (2021)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [Kernel Flows: from learning kernels from data into the abyss](https://mlsanity.com/paper/1808.04475/) (2018)
- [Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network](https://mlsanity.com/paper/2201.02217/) (2022)
- [Nonparametric inference of interaction laws in systems of agents from trajectory data](https://mlsanity.com/paper/1812.06003/) (2018)
- [Solving PDE-constrained Control Problems Using Operator Learning](https://mlsanity.com/paper/2111.04941/) (2021)
- [Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories](https://mlsanity.com/paper/2007.15174/) (2020)
- [Learning dynamical systems from data: a simple cross-validation perspective](https://mlsanity.com/paper/2007.05074/) (2020)
- [Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles](https://mlsanity.com/paper/2010.15694/) (2020)
- [Consistency of Empirical Bayes And Kernel Flow For Hierarchical Parameter Estimation](https://mlsanity.com/paper/2005.11375/) (2020)
- [A data-driven peridynamic continuum model for upscaling molecular dynamics](https://mlsanity.com/paper/2108.04883/) (2021)
- [Data-driven learning of nonlocal models: from high-fidelity simulations to constitutive laws](https://mlsanity.com/paper/2012.04157/) (2020)
- [Numerical methods for nonlocal and fractional models](https://mlsanity.com/paper/2002.01401/) (2020)
- [Identifiability of interaction kernels in mean-field equations of interacting particles](https://mlsanity.com/paper/2106.05565/) (2021)
- [Nonlocal Neural Networks, Nonlocal Diffusion and Nonlocal Modeling](https://mlsanity.com/paper/1806.00681/) (2018)
- [Extracting stochastic dynamical systems with α-stable L'evy noise from data](https://mlsanity.com/paper/2109.14881/) (2021)
- [An optimization-based approach to parameter learning for fractional type nonlocal models](https://mlsanity.com/paper/2010.03666/) (2020)
- [Learning nonlocal regularization operators](https://mlsanity.com/paper/2001.09092/) (2020)

## Built on by (most cited first, 10 of 10)

- [Transformer Meets Boundary Value Inverse Problems](https://mlsanity.com/paper/2209.14977/) (2022)
- [Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model for Complex Material Responses](https://mlsanity.com/paper/2401.06070/) (2024)
- [Nonlocal Attention Operator: Materializing Hidden Knowledge Towards Interpretable Physics Discovery](https://mlsanity.com/paper/2408.07307/) (2024)
- [Bayesian Nonlocal Operator Regression (BNOR): A Data-Driven Learning Framework of Nonlocal Models with Uncertainty Quantification](https://mlsanity.com/paper/2211.01330/) (2022)
- [Towards a unified nonlocal, peridynamics framework for the coarse-graining of molecular dynamics data with fractures](https://mlsanity.com/paper/2301.04540/) (2023)
- [Monotone Peridynamic Neural Operator for Nonlinear Material Modeling with Conditionally Unique Solutions](https://mlsanity.com/paper/2505.01060/) (2025)
- [A Data-Adaptive Prior for Bayesian Learning of Kernels in Operators](https://mlsanity.com/paper/2212.14163/) (2022)
- [Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structures](https://mlsanity.com/paper/2508.01854/) (2025)
- [Small noise analysis for Tikhonov and RKHS regularizations](https://mlsanity.com/paper/2305.11055/) (2023)
- [Embedded Nonlocal Operator Regression (ENOR): Quantifying model error in learning nonlocal operators](https://mlsanity.com/paper/2410.20331/) (2024)

## Abstract (arXiv)

Nonlocal operators with integral kernels have become a popular tool for designing solution maps between function spaces, due to their efficiency in representing long-range dependence and the attractive feature of being resolution-invariant. In this work, we provide a rigorous identifiability analysis and convergence study for the learning of kernels in nonlocal operators. It is found that the kernel learning is an ill-posed or even ill-defined inverse problem, leading to divergent estimators in the presence of modeling errors or measurement noises. To resolve this issue, we propose a nonparametric regression algorithm with a novel data adaptive RKHS Tikhonov regularization method based on the function space of identifiability. The method yields a noisy-robust convergent estimator of the kernel as the data resolution refines, on both synthetic and real-world datasets. In particular, the method successfully learns a homogenized model for the stress wave propagation in a heterogeneous solid, revealing the unknown governing laws from real-world data at microscale. Our regularization method outperforms baseline methods in robustness, generalizability and accuracy.