# Kernel spaces arising from solutions to physical equations

August 12, 2023 — August 12, 2024

functional analysis

Gaussian

generative

Hilbert space

kernel tricks

regression

spatial

stochastic processes

time series

I have little to say here right now but I needed a placeholder to mark the articles on this topics, because their names are not always obvious.

## 1 References

Albert. 2019. “Gaussian Processes for Data Fulfilling Linear Differential Equations.”

*Proceedings*.
Bao, Qian, Liu, et al. 2022. “An Operator Learning Approach via Function-Valued Reproducing Kernel Hilbert Space for Differential Equations.”

Besginow, and Lange-Hegermann. 2024. “Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equations.” In

*Proceedings of the 36th International Conference on Neural Information Processing Systems*. NIPS ’22.
Bolin, and Wallin. 2021. “Efficient Methods for Gaussian Markov Random Fields Under Sparse Linear Constraints.” In

*Advances in Neural Information Processing Systems*.
Cockayne, Oates, Sullivan, et al. 2016. “Probabilistic Numerical Methods for Partial Differential Equations and Bayesian Inverse Problems.”

———, et al. 2017. “Probabilistic Numerical Methods for PDE-Constrained Bayesian Inverse Problems.” In

*AIP Conference Proceedings*.
Cotter, Dashti, and Stuart. 2010. “Approximation of Bayesian Inverse Problems for PDEs.”

*SIAM Journal on Numerical Analysis*.
Gulian, Frankel, and Swiler. 2022. “Gaussian Process Regression Constrained by Boundary Value Problems.”

*Computer Methods in Applied Mechanics and Engineering*.
Harkonen, Lange-Hegermann, and Raita. 2023. “Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients.” In

*Proceedings of the 40th International Conference on Machine Learning*.
Henderson. 2023. “PDE Constrained Kernel Regression Methods.”

Henderson, Noble, and Roustant. 2023. “Characterization of the Second Order Random Fields Subject to Linear Distributional PDE Constraints.”

*Bernoulli*.
Hutchinson, Terenin, Borovitskiy, et al. 2021. “Vector-Valued Gaussian Processes on Riemannian Manifolds via Gauge Independent Projected Kernels.” In

*Advances in Neural Information Processing Systems*.
Kim, Luettgen, Paynabar, et al. 2023. “Physics-Based Penalization for Hyperparameter Estimation in Gaussian Process Regression.”

*Computers & Chemical Engineering*.
Krämer, Schmidt, and Hennig. 2022. “Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations.” In

*Proceedings of The 25th International Conference on Artificial Intelligence and Statistics*.
Kübler, Muandet, and Schölkopf. 2019. “Quantum Mean Embedding of Probability Distributions.”

*Physical Review Research*.
Lange-Hegermann. 2018. “Algorithmic Linearly Constrained Gaussian Processes.” In

*Proceedings of the 32nd International Conference on Neural Information Processing Systems*. NIPS’18.
———. 2021. “Linearly Constrained Gaussian Processes with Boundary Conditions.” In

*Proceedings of the 24th International Conference on Artificial Intelligence and Statistics*. Proceedings of Machine Learning Research.
Magnani, Krämer, Eschenhagen, et al. 2022. “Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs.”

Negiar. 2023. “Constrained Machine Learning: Algorithms and Models.”

Oates, Cockayne, Aykroyd, et al. 2019. “Bayesian Probabilistic Numerical Methods in Time-Dependent State Estimation for Industrial Hydrocyclone Equipment.”

*Journal of the American Statistical Association*.
Ranftl. n.d. “Physics-Consistency of Infinite Neural Networks.”

Sigrist, Künsch, and Stahel. 2015. “Stochastic Partial Differential Equation Based Modelling of Large Space-Time Data Sets.”

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.
Singh, and Principe. 2022. “A Physics Inspired Functional Operator for Model Uncertainty Quantification in the RKHS.”

Stepaniants. n.d. “Learning Partial Differential Equations in Reproducing Kernel Hilbert Spaces.”

Wang, Cockayne, and Oates. 2018. “On the Bayesian Solution of Differential Equations.” In

*38th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering*.
Zhang, Wang, and Nehorai. 2020. “Optimal Transport in Reproducing Kernel Hilbert Spaces: Theory and Applications.”

*IEEE Transactions on Pattern Analysis and Machine Intelligence*.