mlsanityData to 7 Oct 2026

Data adaptive RKHS Tikhonov regularization for learning kernels in operators

arXiv 2203.03791 ↗First posted 8 Mar 2022stat.ML cs.LGcited by 8

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Abstract (arXiv)

We present DARTR: a Data Adaptive RKHS Tikhonov Regularization method for the linear inverse problem of nonparametric learning of function parameters in operators. A key ingredient is a system intrinsic data-adaptive (SIDA) RKHS, whose norm restricts the learning to take place in the function space of identifiability. DARTR utilizes this norm and selects the regularization parameter by the L-curve method. We illustrate its performance in examples including integral operators, nonlinear operators and nonlocal operators with discrete synthetic data. Numerical results show that DARTR leads to an accurate estimator robust to both numerical error due to discrete data and noise in data, and the estimator converges at a consistent rate as the data mesh refines under different levels of noises, outperforming two baseline regularizers using l2 and L2 norms.

Key facts. Cited by 8 papers in the map (data through 7 Oct 2026). Builds on 11 of its 52 references that are in the map. First posted on arXiv 8 Mar 2022.

Builds on

11 of 52 references.

References · 11

  1. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  2. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  3. 2013Early stopping and non-parametric regression: An optimal data-dependent stopping rulecited by 86
  4. 2020DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problemscited by 61
  5. 2015Operator-valued Kernels for Learning from Functional Response Datacited by 59
  6. 2018Kernel Flows: from learning kernels from data into the abysscited by 49
  7. 2020Learning interaction kernels in stochastic systems of interacting particles from multiple trajectoriescited by 24
  8. 2020Learning interaction kernels in mean-field equations of 1st-order systems of interacting particlescited by 21
  9. 2021A data-driven peridynamic continuum model for upscaling molecular dynamicscited by 17
  10. 2020Data-driven learning of nonlocal models: from high-fidelity simulations to constitutive lawscited by 13
  11. 2021Identifiability of interaction kernels in mean-field equations of interacting particlescited by 11

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 2; 2023: 1; 2024: 4; 2025: 0; 2026: 1.

8 papers.

  1. 2024Nonlocal Attention Operator: Materializing Hidden Knowledge Towards Interpretable Physics Discoverycited by 18
  2. 2024Self-test loss functions for learning weak-form operators and gradient flowscited by 4
  3. 2024Learning Memory Kernels in Generalized Langevin Equationscited by 3
  4. 2024Interacting Particle Systems on Networks: joint inference of the network and the interaction kernelcited by 2
  5. 2022A Data-Adaptive Prior for Bayesian Learning of Kernels in Operatorscited by 1
  6. 2022Unsupervised learning of observation functions in state-space models by nonparametric moment methodscited by 0
  7. 2023Small noise analysis for Tikhonov and RKHS regularizationscited by 0
  8. 2026Kernel Regression with Tensor Trains and Hadamard Overparameterizationcited by 0