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<p>We present a Physics-Informed Neural Network framework that combines Chebyshev spectral approximation with Sobolev regularization (CS-PINN) for solving partial differential equations (PDEs). Our approach addresses two fundamental challenges in PDE solution approximation: accuracy of derivative computation and solution regularity. By projecting neural network outputs onto Chebyshev polynomial basis, we replace computationally expensive automatic differentiation with analytical spectral differentiation, enabling efficient and accurate evaluation of higher-order derivatives through established recurrence relations. Simultaneously, we incorporate Sobolev regularization to constrain solutions within appropriate function spaces, directly addressing the spectral bias of neural networks while ensuring solutions maintain the regularity inherent to the underlying physics. When evaluated against conventional PINNs and Sobolev-PINNs across various forward and inverse problems, CS-PINN demonstrates improved accuracy and stability in our experiments, particularly for higher-order PDEs and complex geometries. This work extends the capabilities of data-driven PDE solvers, offering a mathematically rigorous yet computationally efficient alternative that maintains the flexibility of neural network approaches while overcoming their key limitations in derivative computation accuracy, spectral bias, and computational efficiency for higher-order PDEs.</p>

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Chebyshev-Sobolev Physics-Informed Neural Networks for General PDE Solutions

  • 期刊论文

  • 发表日期：

    2025年8月14日

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  • Shikun Chen,
  • Songquan Xiong,
  • Yangguang Liu

摘要

We present a Physics-Informed Neural Network framework that combines Chebyshev spectral approximation with Sobolev regularization (CS-PINN) for solving partial differential equations (PDEs). Our approach addresses two fundamental challenges in PDE solution approximation: accuracy of derivative computation and solution regularity. By projecting neural network outputs onto Chebyshev polynomial basis, we replace computationally expensive automatic differentiation with analytical spectral differentiation, enabling efficient and accurate evaluation of higher-order derivatives through established recurrence relations. Simultaneously, we incorporate Sobolev regularization to constrain solutions within appropriate function spaces, directly addressing the spectral bias of neural networks while ensuring solutions maintain the regularity inherent to the underlying physics. When evaluated against conventional PINNs and Sobolev-PINNs across various forward and inverse problems, CS-PINN demonstrates improved accuracy and stability in our experiments, particularly for higher-order PDEs and complex geometries. This work extends the capabilities of data-driven PDE solvers, offering a mathematically rigorous yet computationally efficient alternative that maintains the flexibility of neural network approaches while overcoming their key limitations in derivative computation accuracy, spectral bias, and computational efficiency for higher-order PDEs.