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<p>As an emerging meshless method for solving partial differential equations (PDEs), physics-informed neural networks (PINNs) have shown great potential in solving forward and inverse problems in fluid mechanics. Although the PINNs have been successfully applied to many branches of applied mathematics in recent years, their effectiveness and accuracy still require further improvement. Beyond the network architecture and learning algorithm, the preprocessing of data has been identified as a critical factor for PINNs performance. This study systematically compares PINNs preprocessing methods, with particular focus on three universal approaches (the non-dimensionalization, inner normalization layers, and synergistic normalization method for data and equation). We evaluate these approaches through ten benchmark fluid mechanics cases governed by fundamental PDEs types: elliptic, parabolic, hyperbolic, and mixed-type, covering forward and inverse problems. The results of the test cases show that adding normalization operations in PINNs significantly improves performance, which enhances training stability. However, their effectiveness varies substantially depending on the type of governing PDEs. For hyperbolic and mixed-type equations, which often exhibit ill-conditioned loss landscapes, the synergistic normalization method outperforms the alternatives notably. In contrast, for elliptic and parabolic problems with a well-conditioned loss, the SynerNorm method may be less effective than the inner normalization layer method. This study provides a practical reference for selecting preprocessing methods in different physical scenarios.</p>

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A comparative study of preprocessing methods for physics-informed neural networks in fluid mechanics

  • 期刊论文

  • 发表日期：

    2026年7月4日

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  • Shengbo Li,
  • Yuanjun Dai,
  • Zhenxu Sun,
  • Dilong Guo,
  • Guowei Yang

摘要

As an emerging meshless method for solving partial differential equations (PDEs), physics-informed neural networks (PINNs) have shown great potential in solving forward and inverse problems in fluid mechanics. Although the PINNs have been successfully applied to many branches of applied mathematics in recent years, their effectiveness and accuracy still require further improvement. Beyond the network architecture and learning algorithm, the preprocessing of data has been identified as a critical factor for PINNs performance. This study systematically compares PINNs preprocessing methods, with particular focus on three universal approaches (the non-dimensionalization, inner normalization layers, and synergistic normalization method for data and equation). We evaluate these approaches through ten benchmark fluid mechanics cases governed by fundamental PDEs types: elliptic, parabolic, hyperbolic, and mixed-type, covering forward and inverse problems. The results of the test cases show that adding normalization operations in PINNs significantly improves performance, which enhances training stability. However, their effectiveness varies substantially depending on the type of governing PDEs. For hyperbolic and mixed-type equations, which often exhibit ill-conditioned loss landscapes, the synergistic normalization method outperforms the alternatives notably. In contrast, for elliptic and parabolic problems with a well-conditioned loss, the SynerNorm method may be less effective than the inner normalization layer method. This study provides a practical reference for selecting preprocessing methods in different physical scenarios.