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The work develops numerical methods of a technology of balanced identification for mathematical models with differential equations (DE). In this case the technology generates inverse problems for DE, the right side of which is a composition of unknown functions. Usually, to approximate these variational problems by finite dimensional mathematical programming problem (MPP) we combine mesh and polynomial approximations respectively for inside and outside functions in composition. A new approach is based on a mesh representation of both functions and on a piecewise linear approximation (PWLA) of the right side function of differential equation. Two methods of PWLA are considered: The 1st way gives MPP with smooth functions, where local optimization solvers are effective, but it is difficult to find a global optimum. The 2nd way gives MPP with special SOS2-constraints, which can be solved by modern global optimization solvers. Both methods are demonstrated for DE with one and two variables.

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Piecewise Linear Approximations in the Balanced Identification of Models with Differential Equations

  • 会议论文

  • 发表日期：

    2025年

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  • Vladimir Voloshinov,
  • Alexander Sokolov

摘要

The work develops numerical methods of a technology of balanced identification for mathematical models with differential equations (DE). In this case the technology generates inverse problems for DE, the right side of which is a composition of unknown functions. Usually, to approximate these variational problems by finite dimensional mathematical programming problem (MPP) we combine mesh and polynomial approximations respectively for inside and outside functions in composition. A new approach is based on a mesh representation of both functions and on a piecewise linear approximation (PWLA) of the right side function of differential equation. Two methods of PWLA are considered: The 1st way gives MPP with smooth functions, where local optimization solvers are effective, but it is difficult to find a global optimum. The 2nd way gives MPP with special SOS2-constraints, which can be solved by modern global optimization solvers. Both methods are demonstrated for DE with one and two variables.