# Physics Informed Deep Kernel Learning

arXiv: https://arxiv.org/abs/2006.04976 · HTML: https://mlsanity.com/paper/2006.04976/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jun 2020.

## Built on by (most cited first, 12 of 12)

- [AutoIP: A United Framework to Integrate Physics into Gaussian Processes](https://mlsanity.com/paper/2202.12316/) (2022)
- [Physics-informed kernel learning](https://mlsanity.com/paper/2409.13786/) (2024)
- [Knowledge-augmented Deep Learning and Its Applications: A Survey](https://mlsanity.com/paper/2212.00017/) (2022)
- [The Hyperdimensional Transform for Distributional Modelling, Regression and Classification](https://mlsanity.com/paper/2311.08150/) (2023)
- [Ensemble Transport Filter via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/2407.11518/) (2024)
- [On Connecting Deep Trigonometric Networks with Deep Gaussian Processes: Covariance, Expressivity, and Neural Tangent Kernel](https://mlsanity.com/paper/2203.07411/) (2022)
- [Learning material synthesis-process-structure-property relationship by data fusion: Bayesian Coregionalization N-Dimensional Piecewise Function Learning](https://mlsanity.com/paper/2311.06228/) (2023)
- [Deep Latent Force Models: ODE-based Process Convolutions for Bayesian Deep Learning](https://mlsanity.com/paper/2311.14828/) (2023)
- [Response to Promises and Pitfalls of Deep Kernel Learning](https://mlsanity.com/paper/2509.21228/) (2025)
- [Multifidelity Gaussian process regression for solving nonlinear partial differential equations](https://mlsanity.com/paper/2605.10383/) (2026)
- [APIC: Amortized Physics-Informed Calibration using Neural Processes](https://mlsanity.com/paper/2606.03355/) (2026)
- [A unified perspective of Gaussian process approximation for differential equations](https://mlsanity.com/paper/2607.06292/) (2026)

## Abstract (arXiv)

Deep kernel learning is a promising combination of deep neural networks and nonparametric function learning. However, as a data driven approach, the performance of deep kernel learning can still be restricted by scarce or insufficient data, especially in extrapolation tasks. To address these limitations, we propose Physics Informed Deep Kernel Learning (PI-DKL) that exploits physics knowledge represented by differential equations with latent sources. Specifically, we use the posterior function sample of the Gaussian process as the surrogate for the solution of the differential equation, and construct a generative component to integrate the equation in a principled Bayesian hybrid framework. For efficient and effective inference, we marginalize out the latent variables in the joint probability and derive a collapsed model evidence lower bound (ELBO), based on which we develop a stochastic model estimation algorithm. Our ELBO can be viewed as a nice, interpretable posterior regularization objective. On synthetic datasets and real-world applications, we show the advantage of our approach in both prediction accuracy and uncertainty quantification.