# On Boundary Conditions Parametrized by Analytic Functions

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Part of the Lecture Notes in Computer Science book series (LNCS,volume 13366)

## Abstract

Computer algebra can answer various questions about partial differential equations using symbolic algorithms. However, the inclusion of data into equations is rare in computer algebra. Therefore, recently, computer algebra models have been combined with Gaussian processes, a regression model in machine learning, to describe the behavior of certain differential equations under data. While it was possible to describe polynomial boundary conditions in this context, we extend these models to analytic boundary conditions. Additionally, we describe the necessary algorithms for Gröbner and Janet bases of Weyl algebras with certain analytic coefficients. Using these algorithms, we provide examples of divergence-free flow in domains bounded by analytic functions and adapted to observations.

### Keywords

• Gaussian processes
• Boundary conditions
• Gröbner bases
• Partial differential equations

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1. 1.

The function k is positive (semi)definite if and only if for any $$x_1,\ldots ,x_n\in D$$ the matrix $$K=(k(x_i,x_j))_{i,j}\in {\mathbb {R}}^{n\ell \times n\ell }$$ is positive (semi)definite, i.e., $$K \succeq 0$$.

2. 2.

For $${{\,\mathrm{\mathcal{G}\mathcal{P}}\,}}(0,k)$$, the set $${\mathcal {H}}^0(g)$$ generated as a vector space by the $$x\mapsto k(x_i,x)$$ for $$x_i\in D$$ with scalar product $$\left\langle k(x_i,-),k(x_j,-) \right\rangle := k(x_i,x_j)$$ is a pre-Hilbert space. Its closure $${\mathcal {H}}(g)$$ is the reproducing kernel Hilbert space of the GP g .

3. 3.

In algebraic system theory, one usually works with injective cogenerators $$\mathcal {F}$$ . Injective cogenerators allow to infer back from analysis in $$\mathcal {F}$$ to algebra over $$R$$. In our setting, this step back is superfluous, as the algebra cannot encode data points.

4. 4.

Of course, the constructions in this and the following section work over any sufficiently algorithmic differential field of characteristic zero, not only $${\mathbb {R}}$$. In practice, we assume to work over a computable subfield of $${\mathbb {R}}$$.

5. 5.

$$B_3'$$ is obtained as the product of the standard deviations obtained by conditioning d one-dimensional squared exponential covariances to the data points (0, 0) and (1, 0).

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## Acknowledgment

The authors thank Andreas Besginow for discussions and the anonymous reviewers for helpful comments, both of which improved the contents of this paper.

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Correspondence to Markus Lange-Hegermann .

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