Skip to main content
Book cover

International Workshop on Computer Algebra in Scientific Computing

CASC 2022: Computer Algebra in Scientific Computing pp 225–245Cite as

On Boundary Conditions Parametrized by Analytic Functions

  • 282 Accesses

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

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-031-14788-3_13
  • Chapter length: 21 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   59.99
Price excludes VAT (USA)
  • ISBN: 978-3-031-14788-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   79.99
Price excludes VAT (USA)
Learn about institutional subscriptions
Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.

Notes

  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 [2].

  3. 3.

    In algebraic system theory, one usually works with injective cogenerators \(\mathcal {F}\) [32]. 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).

References

  1. Bächler, T., Gerdt, V., Lange-Hegermann, M., Robertz, D.: Algorithmic Thomas decomposition of algebraic and differential systems. J. Symb. Comput. 47(10), 1233–1266 (2012)

    CrossRef  MathSciNet  Google Scholar 

  2. Berlinet, A., Thomas-Agnan, C.: Reproducing Kernel Hilbert Spaces in Probability and Statistics. Kluwer, Boston (2004)

    CrossRef  Google Scholar 

  3. Berns, F., Lange-Hegermann, M., Beecks, C.: Towards Gaussian processes for automatic and interpretable anomaly detection in industry 4.0. In: Proceedings of the International Conference on Innovative Intelligent Industrial Production and Logistics - IN4PL (2020)

    Google Scholar 

  4. Bogachev, V.I.: Gaussian Measures. Mathematical Surveys and Monographs, vol. 62. American Mathematical Society, Providence (1998)

    Google Scholar 

  5. Chyzak, F., Quadrat, A., Robertz, D.: Effective algorithms for parametrizing linear control systems over Ore algebras. Appl. Algebra Eng. Commun. Comput. 16(5), 319–376 (2005)

    CrossRef  MathSciNet  Google Scholar 

  6. Cramér, H., Leadbetter, M.R.: Stationary and Related Stochastic Processes. Dover Publications Inc., Mineola (2004)

    MATH  Google Scholar 

  7. Ehrenpreis, L.: Solution of some problems of division. I. Division by a polynomial of derivation. Am. J. Math. 76, 883–903 (1954)

    Google Scholar 

  8. Gerdt, V.P., Lange-Hegermann, M., Robertz, D.: The MAPLE package TDDS for computing Thomas decompositions of systems of nonlinear PDEs. Comput. Phys. Commun. 234, 202–215 (2019)

    CrossRef  MathSciNet  Google Scholar 

  9. Graepel, T.: Solving noisy linear operator equations by Gaussian processes: application to ordinary and partial differential equations. In: Proceedings of the Twentieth International Conference on International Conference on Machine Learning, pp. 234–241 (2003)

    Google Scholar 

  10. Gulian, M., Frankel, A., Swiler, L.: Gaussian process regression constrained by boundary value problems. Comput. Methods Appl. Mech. Eng. 388, Paper No. 114117, 18 (2022)

    Google Scholar 

  11. Honkela, A., et al.: Genome-wide modeling of transcription kinetics reveals patterns of RNA production delays. In: Proceedings of the National Academy of Sciences (2015)

    Google Scholar 

  12. Jacobson, N.: The Theory of Rings. American Mathematical Society Mathematical Surveys, vol. II. American Mathematical Society (1943)

    Google Scholar 

  13. Jaynes, E.T.: Probability Theory. Cambridge University Press, Cambridge (2003)

    CrossRef  Google Scholar 

  14. Jidling, C., et al.: Probabilistic modelling and reconstruction of strain. Nucl. Instrum. Methods Phys. Res. Sect. B Beam Interact. Mater. Atoms 436, 141–155 (2018)

    Google Scholar 

  15. Jidling, C., Wahlström, N., Wills, A., Schön, T.B.: Linearly constrained gaussian processes. In: Advances in Neural Information Processing Systems (2017)

    Google Scholar 

  16. John, D., Heuveline, V., Schober, M.: GOODE: a Gaussian off-the-shelf ordinary differential equation solver. In: Proceedings of the 36th International Conference on Machine Learning (2019)

    Google Scholar 

  17. Lange-Hegermann, M.: Counting solutions of differential equations. Ph.D. thesis, RWTH Aachen (2014)

    Google Scholar 

  18. Lange-Hegermann, M.: The differential dimension polynomial for characterizable differential ideals. In: Böckle, G., Decker, W., Malle, G. (eds.) Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory, pp. 443–453. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-70566-8_18

    CrossRef  Google Scholar 

  19. Lange-Hegermann, M.: Algorithmic linearly constrained Gaussian processes. In: Advances in Neural Information Processing Systems (2018)

    Google Scholar 

  20. Lange-Hegermann, M.: The differential counting polynomial. Found. Comput. Math. 18(2), 291–308 (2018)

    CrossRef  MathSciNet  Google Scholar 

  21. Lange-Hegermann, M.: Linearly constrained Gaussian processes with boundary conditions. In: International Conference on Artificial Intelligence and Statistics. PMLR (2021)

    Google Scholar 

  22. Lange-Hegermann, M., Robertz, D.: Thomas decompositions of parametric nonlinear control systems. IFAC Proc. Vol. 46(2), 296–301 (2013)

    Google Scholar 

  23. Lange-Hegermann, M., Robertz, D.: Thomas decomposition and nonlinear control systems. In: Quadrat, A., Zerz, E. (eds.) Algebraic and Symbolic Computation Methods in Dynamical Systems. ADD, vol. 9, pp. 117–146. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-38356-5_4

    CrossRef  MATH  Google Scholar 

  24. Lange-Hegermann, M., Robertz, D., Seiler, W.M., Seiß, M.: Singularities of algebraic differential equations. Adv. Appl. Math. 131, Paper No. 102266, 56 (2021)

    Google Scholar 

  25. Macêdo, I., Castro, R.: Learning divergence-free and curl-free vector fields with matrix-valued kernels. Instituto Nacional de Matematica Pura e Aplicada, Brasil, Technical report (2008)

    Google Scholar 

  26. Malgrange, B.: Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution (1955/56). http://aif.cedram.org/item?id=AIF_1955__6__271_0

  27. Malgrange, B.: Division des distributions. Séminaire Schwartz (1959)

    Google Scholar 

  28. McConnell, J.C., Robson, J.C.: Noncommutative Noetherian Rings. Graduate Studies in Mathematics, vol. 30. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  29. Neal, R.M.: Priors for infinite networks. In: Neal, R.M. (ed.) Bayesian Learning for Neural Networks, pp. 29–53. Springer, New York (1996). https://doi.org/10.1007/978-1-4612-0745-0_2

    CrossRef  MATH  Google Scholar 

  30. Nicholson, J., Kiessler, P., Brown, D.A.: A kernel-based approach for modelling Gaussian processes with functional information. arXiv preprint arXiv:2201.11023 (2022)

  31. Oberst, U.: Multidimensional constant linear systems. Acta Appl. Math. 20(1–2), 1–175 (1990)

    CrossRef  MathSciNet  Google Scholar 

  32. Quadrat, A.: Systèmes et Structures - Une approche de la théorie mathématique des systèmes par l’analyse algébrique constructive. Habilitation thesis, Université de Nice (Sophia Antipolis), France, April 2010

    Google Scholar 

  33. Quadrat, A.: Grade filtration of linear functional systems. Acta Appl. Math. 127, 27–86 (2013)

    CrossRef  MathSciNet  Google Scholar 

  34. Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006)

    MATH  Google Scholar 

  35. Regensburger, G., Rosenkranz, M., Middeke, J.: A skew polynomial approach to integro-differential operators. In: ISSAC 2009–Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation, pp. 287–294. ACM, New York (2009)

    Google Scholar 

  36. Robertz, D.: Formal Algorithmic Elimination for PDEs. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11445-3

    CrossRef  MATH  Google Scholar 

  37. Robertz, D.: Recent progress in an algebraic analysis approach to linear systems. Multidimension. Syst. Signal Process. 26(2), 349–388 (2014). https://doi.org/10.1007/s11045-014-0280-9

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. Rosenkranz, M., Regensburger, G.: Solving and factoring boundary problems for linear ordinary differential equations in differential algebras. J. Symb. Comput. 43(8), 515–544 (2008)

    CrossRef  MathSciNet  Google Scholar 

  39. Särkkä, S.: Linear operators and stochastic partial differential equations in Gaussian process regression. In: Honkela, T., Duch, W., Girolami, M., Kaski, S. (eds.) ICANN 2011. LNCS, vol. 6792, pp. 151–158. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-21738-8_20

    CrossRef  Google Scholar 

  40. Scheuerer, M., Schlather, M.: Covariance models for divergence-free and curl-free random vector fields. Stoch. Models 28(3), 433–451 (2012)

    CrossRef  MathSciNet  Google Scholar 

  41. Simon-Gabriel, C.J., Schölkopf, B.: Kernel distribution embeddings: universal kernels, characteristic kernels and kernel metrics on distributions. J. Mach. Learn. Res. 19, Paper No. 44, 29 (2018)

    Google Scholar 

  42. Solin, A., Kok, M.: Know your boundaries: constraining Gaussian processes by variational harmonic features. In: Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (2019)

    Google Scholar 

  43. Solin, A., Kok, M., Wahlström, N., Schön, T.B., Särkkä, S.: Modeling and interpolation of the ambient magnetic field by Gaussian processes. IEEE Trans. Robot. 34(4), 1112–1127 (2018)

    CrossRef  Google Scholar 

  44. Tan, M.H.Y.: Gaussian process modeling with boundary information. Statist. Sinica 28(2), 621–648 (2018)

    MathSciNet  MATH  Google Scholar 

  45. Thewes, S., Lange-Hegermann, M., Reuber, C., Beck, R.: Advanced Gaussian process modeling techniques. In: Design of Experiments (DoE) in Powertrain Development (2015)

    Google Scholar 

  46. Tougeron, J.C.: Idéaux de fonctions différentiables. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer, Heidelberg (1972). https://doi.org/10.1007/978-3-662-59320-2

    CrossRef  MATH  Google Scholar 

  47. Wahlström, N., Kok, M., Schön, T.B., Gustafsson, F.: Modeling magnetic fields using Gaussian processes. In: Proceedings of the 38th International Conference on Acoustics, Speech, and Signal Processing (ICASSP) (2013)

    Google Scholar 

  48. Whitney, H.: On ideals of differentiable functions. Am. J. Math. 70, 635–658 (1948)

    CrossRef  MathSciNet  Google Scholar 

  49. Zerz, E., Seiler, W.M., Hausdorf, M.: On the inverse syzygy problem. Commun. Algebra 38(6), 2037–2047 (2010)

    CrossRef  MathSciNet  Google Scholar 

  50. Zimmer, C., Meister, M., Nguyen-Tuong, D.: Safe active learning for time-series modeling with Gaussian processes. In: Advances in Neural Information Processing Systems (2018)

    Google Scholar 

Download references

Acknowledgment

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

Author information

Authors and Affiliations

  1. Technische Hochschule Ostwestfalen-Lippe, inIT (Institute Industrial IT), Lemgo, Germany

    Markus Lange-Hegermann

  2. Lehrstuhl für Algebra und Zahlentheorie, RWTH Aachen University, Aachen, Germany

    Daniel Robertz

Authors
  1. Markus Lange-Hegermann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Robertz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Markus Lange-Hegermann .

Editor information

Editors and Affiliations

  1. Université de Lille, Villeneuve d'Ascq, France

    Prof. François Boulier

  2. Coventry University, Coventry, UK

    Dr. Matthew England

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. Institute of Theoretical and Applied Mechanics, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

Rights and permissions

Reprints and Permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Lange-Hegermann, M., Robertz, D. (2022). On Boundary Conditions Parametrized by Analytic Functions. In: Boulier, F., England, M., Sadykov, T.M., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2022. Lecture Notes in Computer Science, vol 13366. Springer, Cham. https://doi.org/10.1007/978-3-031-14788-3_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-14788-3_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-14787-6

  • Online ISBN: 978-3-031-14788-3

  • eBook Packages: Computer ScienceComputer Science (R0)