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Gaussian Processes for Data Fulfilling Linear Differential Equations ^{ †}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. GP Regression for Data from Linear PDEs

#### 2.1. Construction of Kernels for PDEs

#### 2.2. Linear Modeling of Sources

## 3. Application Cases

#### 3.1. Laplace’s Equation in Two Dimensions

#### 3.2. Helmholtz Equation: Source and Wavenumber Reconstruction

#### 3.3. Heat Equation

## 4. Summary and Outlook

## Acknowledgments

## References

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**Figure 1.**Analytical solution of Laplace equation (

**top left**) and GP reconstruction with source-free Mercer kernel (26) (

**top right**) with absolute error (

**bottom left**) and predicted 95% confidence interval (

**bottom right**). Sources lie outside the black square region and measurement positions are marked by black dots.

**Figure 2.**GP reconstruction of case in Figure 1 with generic squared exponential kernel (

**top left**) with predicted 95% confidence interval (

**bottom left**). Difference to reconstruction with source-free kernel (26) (

**top right**) and source density $\overline{q}=\Delta \overline{u}$ of prediction (

**bottom right**).

**Figure 3.**Reconstruction error for Helmholtz equation with different sensor count (top, bottom left) and reconstructed source strengths $\mathbf{q}$ with $95\%$ confidence interval according to posterior (18) and (19). Negative log likelihood (bottom right) with optimum at ${k}_{0}^{\mathrm{ML}}=9.19$ for Bessel kernel [11] (solid line), whereas the actual value (dotted line) is ${k}_{0}=9.16$. The length scale of a squared exponential kernel (dashed line) is less peaked.

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**MDPI and ACS Style**

Albert, C.G.
Gaussian Processes for Data Fulfilling Linear Differential Equations . *Proceedings* **2019**, *33*, 5.
https://doi.org/10.3390/proceedings2019033005

**AMA Style**

Albert CG.
Gaussian Processes for Data Fulfilling Linear Differential Equations . *Proceedings*. 2019; 33(1):5.
https://doi.org/10.3390/proceedings2019033005

**Chicago/Turabian Style**

Albert, Christopher G.
2019. "Gaussian Processes for Data Fulfilling Linear Differential Equations " *Proceedings* 33, no. 1: 5.
https://doi.org/10.3390/proceedings2019033005