# Gaussian Processes for Data Fulfilling Linear Differential Equations

@article{Albert2019GaussianPF, title={Gaussian Processes for Data Fulfilling Linear Differential Equations}, author={Christopher G. Albert}, journal={Proceedings}, year={2019} }

A method to reconstruct fields, source strengths and physical parameters based on Gaussian process regression is presented for the case where data are known to fulfill a given linear differential equation with localized sources. The approach is applicable to a wide range of data from physical measurements and numerical simulations. It is based on the well-known invariance of the Gaussian under linear operators, in particular differentiation. Instead of using a generic covariance function to…

## 7 Citations

### Gaussian Process Regression for Data Fulfilling Linear Differential Equations with Localized Sources †

- Mathematics, Computer ScienceEntropy
- 2020

Specialized Gaussian process regression is presented for data that are known to fulfill a given linear differential equation with vanishing or localized sources, and it generates only physically possible solutions, and estimated hyperparameters represent physical properties.

### Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients

- Computer ScienceArXiv
- 2022

The Ehrenpreis-Palamodov fundamental principle is applied, which works like a non-linear Fourier transform, to construct GP kernels mirroring standard spectral methods for GPs, and can infer probable solutions of linear PDE systems from any data such as noisy measurements, or pointwise deﬁned initial and boundary conditions.

### Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers

- Computer ScienceArXiv
- 2022

The results enable the seamless integration of mechanistic models as modular building blocks into probabilistic models by blurring the boundaries between numerical analysis and Bayesian inference.

### Bayesian Uncertainty Quantification with Multi-Fidelity Data and Gaussian Processes for Impedance Cardiography of Aortic Dissection

- Computer ScienceEntropy
- 2020

The fully Bayesian uncertainty quantification method is devised in a notation following the tradition of E.T. Jaynes and found that generalization to an arbitrary number of levels of fidelity and parallelisation becomes rather easy.

### A connection between probability, physics and neural networks

- Computer ScienceMaxEnt 2022
- 2022

The central limit theorem suggests that NNs can be constructed to obey a physical law by choosing the activation functions such that they match a particular kernel in the inﬁnite-width limit.

### InfPolyn, a Nonparametric Bayesian Characterization for Composition-Dependent Interdiffusion Coefficients

- MathematicsMaterials
- 2021

The proposed InfPolyn, Infinite Polynomial, a novel statistical framework to characterize the component-dependent interdiffusion coefficients is proposed, a generalization of the commonly used polynomial fitting method with extended model capacity and flexibility and it is combined with the numerical inversion-based Boltzmann–Matano method for the interDiffusion coefficient estimations.

### Expectation-Maximization Algorithm for the Calibration of Complex Simulator Using a Gaussian Process Emulator

- Computer ScienceEntropy
- 2021

Both the variance and bias of the estimates obtained from the proposed EM algorithm are smaller than those from the existing calibration methods.

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