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Verified Numerical Methods for Ordinary Differential Equations

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Software Verification and Formal Methods for ML-Enabled Autonomous Systems (NSV 2022, FoMLAS 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13466))

Abstract

Ordinary differential equations (ODEs) are used to model the evolution of the state of a system over time. They are ubiquitous in the physical sciences and are often used in computational models with safety-critical applications. For critical computations, numerical solvers for ODEs that provide useful guarantees of their accuracy and correctness are required, but do not always exist in practice. In this work, we demonstrate how to use the Coq proof assistant to verify that a C program correctly and accurately finds the solution to an ODE initial value problem (IVP). Our verification framework is modular, and concisely disentangles the high-level mathematical properties expected of the system being modeled from the low-level behavior of a particular C program. Our approach relies on the construction of two simple functional models in Coq: a floating-point valued functional model for analyzing the intermediate-level behavior of the program, and a real-valued functional model for analyzing the high-level mathematical properties of the system being modeled by the IVP. Our final result is a proof that the floating-point solution returned by the C program is an accurate solution to the IVP, with a good quantitative bound. Our framework assumes only the operational semantics of C and of IEEE-754 floating point arithmetic.

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Notes

  1. 1.

    This particular model problem admits an analytical solution and is therefore not expected to be of practical interest on its own. Instead, it is chosen for demonstrating and analyzing the performance of our logical framework.

  2. 2.

    The form name (arguments) : type := term in Coq binds name to the value of the term of type type; is the type of well-formed propositions.

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Acknowledgments

This work benefited substantially from discussions with David Bindel. We thank Michael Soegtrop for his close reading and helpful feedback. Ariel Kellison is supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Department of Energy Computational Science Graduate Fellowship under Award Number DE-SC0021110.

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Authors and Affiliations

  1. Cornell University, Ithaca, NY, USA

    Ariel E. Kellison

  2. Princeton University, Princeton, NJ, USA

    Andrew W. Appel

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  1. Ariel E. Kellison
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Correspondence to Ariel E. Kellison .

Editor information

Editors and Affiliations

  1. The Hebrew University of Jerusalem, Jerusalem, Israel

    Omri Isac

  2. Rensselaer Polytechnic Institute, New York, NY, USA

    Radoslav Ivanov

  3. The Hebrew University of Jerusalem, Jerusalem, Israel

    Guy Katz

  4. Palo Alto, CA, USA

    Nina Narodytska

  5. University of Trieste, Trieste, Italy

    Laura Nenzi

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Kellison, A.E., Appel, A.W. (2022). Verified Numerical Methods for Ordinary Differential Equations. In: Isac, O., Ivanov, R., Katz, G., Narodytska, N., Nenzi, L. (eds) Software Verification and Formal Methods for ML-Enabled Autonomous Systems. NSV FoMLAS 2022 2022. Lecture Notes in Computer Science, vol 13466. Springer, Cham. https://doi.org/10.1007/978-3-031-21222-2_9

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  • DOI: https://doi.org/10.1007/978-3-031-21222-2_9

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  • Online ISBN: 978-3-031-21222-2

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