Taylor mapping method for solving and learning of dynamic processes

IF 1.1 4区 工程技术 Q3 ENGINEERING, MULTIDISCIPLINARY Inverse Problems in Science and Engineering Pub Date : 2021-09-30 DOI:10.1080/17415977.2021.1977294
U. Iben, C. Wagner
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引用次数: 2

Abstract

In this paper, we discuss the so-called Taylor map approach to solve systems of ordinary differential equations. We demonstrate its capabilities of solving the corresponding inverse problems including parameter identification. The method applies even if the underlying ordinary differential equation is not explicitly known. This procedure is interpreted in terms of Polynomial Neural Networks. Physical knowledge is incorporated into the neural network since its architecture is designed directly on top of the Taylor map approach. This does not only improve the data efficiency but also reduce the effort of the included training procedure. We prove an asymptotic convergence result of polynomial neural networks. On this basis, we demonstrate validation examples to highlight the capabilities of this method in practice.
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求解和学习动态过程的泰勒映射方法
本文讨论了求解常微分方程组的泰勒映射方法。我们证明了它的能力,解决相应的逆问题,包括参数辨识。这种方法即使在基本的常微分方程不明确已知的情况下也适用。这个程序是解释在多项式神经网络方面。物理知识被整合到神经网络中,因为它的架构是直接在泰勒地图方法的基础上设计的。这不仅提高了数据效率,而且减少了包含训练过程的工作量。证明了多项式神经网络的一个渐近收敛结果。在此基础上,我们演示了验证示例,以突出该方法在实践中的能力。
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来源期刊
Inverse Problems in Science and Engineering
Inverse Problems in Science and Engineering 工程技术-工程:综合
自引率
0.00%
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0
审稿时长
6 months
期刊介绍: Inverse Problems in Science and Engineering provides an international forum for the discussion of conceptual ideas and methods for the practical solution of applied inverse problems. The Journal aims to address the needs of practising engineers, mathematicians and researchers and to serve as a focal point for the quick communication of ideas. Papers must provide several non-trivial examples of practical applications. Multidisciplinary applied papers are particularly welcome. Topics include: -Shape design: determination of shape, size and location of domains (shape identification or optimization in acoustics, aerodynamics, electromagnets, etc; detection of voids and cracks). -Material properties: determination of physical properties of media. -Boundary values/initial values: identification of the proper boundary conditions and/or initial conditions (tomographic problems involving X-rays, ultrasonics, optics, thermal sources etc; determination of thermal, stress/strain, electromagnetic, fluid flow etc. boundary conditions on inaccessible boundaries; determination of initial chemical composition, etc.). -Forces and sources: determination of the unknown external forces or inputs acting on a domain (structural dynamic modification and reconstruction) and internal concentrated and distributed sources/sinks (sources of heat, noise, electromagnetic radiation, etc.). -Governing equations: inference of analytic forms of partial and/or integral equations governing the variation of measured field quantities.
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