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<p>Parametric differential equations underpin model prediction and system design across numerous fields from fluid mechanics to structural dynamics. However, traditional solution paradigms require complete high-fidelity numerical simulations for each new parameter set, leading to exponentially growing computational burden in multi-parameter exploration scenarios. This paper proposes an efficient solution framework based on data dimensionality reduction and radial basis functions, establishing an offline-online computational decomposition architecture that achieves spatiotemporal redistribution of computational resources and real-time prediction of parametric differential equations. This architecture completely migrates computationally intensive tasks to the offline stage, including high-fidelity numerical simulations, orthogonal basis extraction, and radial basis network training, while the online stage only requires low-dimensional matrix operations, significantly reducing computational complexity and fundamentally improving the efficiency bottleneck of traditional “point-by-point solution” paradigms. Building upon this architecture, this paper further develops a coupled method of regional clustering dimensionality reduction and adaptive radial basis functions. Through intelligent parameter space partitioning strategy, dedicated POD bases are independently constructed for each local region, and specialized radial basis function networks are trained in the low-dimensional space corresponding to each local POD basis, forming a dual localization modeling strategy of “local POD basis &amp; local RBF network”. Meanwhile, a condition number-controlled automatic optimization algorithm for shape parameters is developed to ensure numerical stability of each local RBF network. Through systematic validation on five typical dynamical systems (Allen-Cahn equation, Burgers equation, Duffing oscillator, Van der Pol oscillator, and Kovasznay flow), experiments demonstrate that this framework achieves significant computational efficiency improvements (18%-96% reduction in computation time) while maintaining relative errors ranging from 0.5% to 7%, providing an effective solution for computationally intensive applications such as real-time control, parameter optimization, and uncertainty quantification.</p>

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Offline-online computational decomposition: an efficient framework for parametric dynamical systems via regional clustering dimensionality reduction and adaptive radial basis functions

  • 期刊论文

  • 发表日期：

    2025年7月21日

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  • Sheng Zhou,
  • Xiong Xiong,
  • Kang Lu,
  • Zheng Zeng,
  • Rongchun Hu

摘要

Parametric differential equations underpin model prediction and system design across numerous fields from fluid mechanics to structural dynamics. However, traditional solution paradigms require complete high-fidelity numerical simulations for each new parameter set, leading to exponentially growing computational burden in multi-parameter exploration scenarios. This paper proposes an efficient solution framework based on data dimensionality reduction and radial basis functions, establishing an offline-online computational decomposition architecture that achieves spatiotemporal redistribution of computational resources and real-time prediction of parametric differential equations. This architecture completely migrates computationally intensive tasks to the offline stage, including high-fidelity numerical simulations, orthogonal basis extraction, and radial basis network training, while the online stage only requires low-dimensional matrix operations, significantly reducing computational complexity and fundamentally improving the efficiency bottleneck of traditional “point-by-point solution” paradigms. Building upon this architecture, this paper further develops a coupled method of regional clustering dimensionality reduction and adaptive radial basis functions. Through intelligent parameter space partitioning strategy, dedicated POD bases are independently constructed for each local region, and specialized radial basis function networks are trained in the low-dimensional space corresponding to each local POD basis, forming a dual localization modeling strategy of “local POD basis & local RBF network”. Meanwhile, a condition number-controlled automatic optimization algorithm for shape parameters is developed to ensure numerical stability of each local RBF network. Through systematic validation on five typical dynamical systems (Allen-Cahn equation, Burgers equation, Duffing oscillator, Van der Pol oscillator, and Kovasznay flow), experiments demonstrate that this framework achieves significant computational efficiency improvements (18%-96% reduction in computation time) while maintaining relative errors ranging from 0.5% to 7%, providing an effective solution for computationally intensive applications such as real-time control, parameter optimization, and uncertainty quantification.