# A Data-Adaptive Prior for Bayesian Learning of Kernels in Operators

arXiv: https://arxiv.org/abs/2212.14163 · HTML: https://mlsanity.com/paper/2212.14163/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 68 references that are in the map.
- First posted on arXiv 29 Dec 2022.

## Builds on (most cited first, 18 of 18)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [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: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [To understand deep learning we need to understand kernel learning](https://mlsanity.com/paper/1802.01396/) (2018)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Kernel Flows: from learning kernels from data into the abyss](https://mlsanity.com/paper/1808.04475/) (2018)
- [Nonparametric inference of interaction laws in systems of agents from trajectory data](https://mlsanity.com/paper/1812.06003/) (2018)
- [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021)
- [Learning interaction kernels in heterogeneous systems of agents from multiple trajectories](https://mlsanity.com/paper/1910.04832/) (2019)
- [Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories](https://mlsanity.com/paper/2007.15174/) (2020)
- [Optimal low-rank approximations of Bayesian linear inverse problems](https://mlsanity.com/paper/1407.3463/) (2014)
- [Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles](https://mlsanity.com/paper/2010.15694/) (2020)
- [A data-driven peridynamic continuum model for upscaling molecular dynamics](https://mlsanity.com/paper/2108.04883/) (2021)
- [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)
- [Nonparametric learning of kernels in nonlocal operators](https://mlsanity.com/paper/2205.11006/) (2022)
- [Data adaptive RKHS Tikhonov regularization for learning kernels in operators](https://mlsanity.com/paper/2203.03791/) (2022)
- [Learning Swarm Interaction Dynamics from Density Evolution](https://mlsanity.com/paper/2112.02675/) (2021)

## Built on by (most cited first, 1 of 1)

- [Small noise analysis for Tikhonov and RKHS regularizations](https://mlsanity.com/paper/2305.11055/) (2023)

## Abstract (arXiv)

Kernels are efficient in representing nonlocal dependence and they are widely used to design operators between function spaces. Thus, learning kernels in operators from data is an inverse problem of general interest. Due to the nonlocal dependence, the inverse problem can be severely ill-posed with a data-dependent singular inversion operator. The Bayesian approach overcomes the ill-posedness through a non-degenerate prior. However, a fixed non-degenerate prior leads to a divergent posterior mean when the observation noise becomes small, if the data induces a perturbation in the eigenspace of zero eigenvalues of the inversion operator. We introduce a data-adaptive prior to achieve a stable posterior whose mean always has a small noise limit. The data-adaptive prior's covariance is the inversion operator with a hyper-parameter selected adaptive to data by the L-curve method. Furthermore, we provide a detailed analysis on the computational practice of the data-adaptive prior, and demonstrate it on Toeplitz matrices and integral operators. Numerical tests show that a fixed prior can lead to a divergent posterior mean in the presence of any of the four types of errors: discretization error, model error, partial observation and wrong noise assumption. In contrast, the data-adaptive prior always attains posterior means with small noise limits.