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Physics-Informed Chebyshev Polynomial Neural Operator for Parametric Partial Differential Equations

Biao Chen, Jing Wang, Hairun Xie, Qineng Wang, Shuai Zhang, Yifan Xia, Jifa Zhang

TL;DR

The paper tackles the challenge of learning parametric PDE solution operators under physics-informed supervision, where conventional MLP-based neural operators suffer from spectral bias and training instability. It introduces the Physics-Informed Chebyshev Polynomial Neural Operator (CPNO), a mesh-free architecture that encodes inputs with Chebyshev spectral features and employs a parameter-driven modulation to build high-order polynomial representations. Theoretical analysis demonstrates spectral convergence and favorable conditioning of the Chebyshev basis, while extensive experiments on Burgers, Allen-Cahn, diffusion-reaction, Navier-Stokes, and transonic airfoil problems show superior accuracy, faster convergence, and robustness to hyperparameters compared with baselines. The results suggest that Chebyshev-based spectral encoding provides a principled, stable foundation for learning parametric PDE solution operators with physics-informed losses, enabling efficient, reliable predictions in complex multi-scale settings.

Abstract

Neural operators have emerged as powerful deep learning frameworks for approximating solution operators of parameterized partial differential equations (PDE). However, current methods predominantly rely on multilayer perceptrons (MLPs) for mapping inputs to solutions, which impairs training robustness in physics-informed settings due to inherent spectral biases and fixed activation functions. To overcome the architectural limitations, we introduce the Physics-Informed Chebyshev Polynomial Neural Operator (CPNO), a novel mesh-free framework that leverages a basis transformation to replace unstable monomial expansions with the numerically stable Chebyshev spectral basis. By integrating parameter dependent modulation mechanism to main net, CPNO constructs PDE solutions in a near-optimal functional space, decoupling the model from MLP-specific constraints and enhancing multi-scale representation. Theoretical analysis demonstrates the Chebyshev basis's near-minimax uniform approximation properties and superior conditioning, with Lebesgue constants growing logarithmically with degree, thereby mitigating spectral bias and ensuring stable gradient flow during optimization. Numerical experiments on benchmark parameterized PDEs show that CPNO achieves superior accuracy, faster convergence, and enhanced robustness to hyperparameters. The experiment of transonic airfoil flow has demonstrated the capability of CPNO in characterizing complex geometric problems.

Physics-Informed Chebyshev Polynomial Neural Operator for Parametric Partial Differential Equations

TL;DR

The paper tackles the challenge of learning parametric PDE solution operators under physics-informed supervision, where conventional MLP-based neural operators suffer from spectral bias and training instability. It introduces the Physics-Informed Chebyshev Polynomial Neural Operator (CPNO), a mesh-free architecture that encodes inputs with Chebyshev spectral features and employs a parameter-driven modulation to build high-order polynomial representations. Theoretical analysis demonstrates spectral convergence and favorable conditioning of the Chebyshev basis, while extensive experiments on Burgers, Allen-Cahn, diffusion-reaction, Navier-Stokes, and transonic airfoil problems show superior accuracy, faster convergence, and robustness to hyperparameters compared with baselines. The results suggest that Chebyshev-based spectral encoding provides a principled, stable foundation for learning parametric PDE solution operators with physics-informed losses, enabling efficient, reliable predictions in complex multi-scale settings.

Abstract

Neural operators have emerged as powerful deep learning frameworks for approximating solution operators of parameterized partial differential equations (PDE). However, current methods predominantly rely on multilayer perceptrons (MLPs) for mapping inputs to solutions, which impairs training robustness in physics-informed settings due to inherent spectral biases and fixed activation functions. To overcome the architectural limitations, we introduce the Physics-Informed Chebyshev Polynomial Neural Operator (CPNO), a novel mesh-free framework that leverages a basis transformation to replace unstable monomial expansions with the numerically stable Chebyshev spectral basis. By integrating parameter dependent modulation mechanism to main net, CPNO constructs PDE solutions in a near-optimal functional space, decoupling the model from MLP-specific constraints and enhancing multi-scale representation. Theoretical analysis demonstrates the Chebyshev basis's near-minimax uniform approximation properties and superior conditioning, with Lebesgue constants growing logarithmically with degree, thereby mitigating spectral bias and ensuring stable gradient flow during optimization. Numerical experiments on benchmark parameterized PDEs show that CPNO achieves superior accuracy, faster convergence, and enhanced robustness to hyperparameters. The experiment of transonic airfoil flow has demonstrated the capability of CPNO in characterizing complex geometric problems.
Paper Structure (25 sections, 3 theorems, 51 equations, 15 figures, 2 tables, 1 algorithm)

This paper contains 25 sections, 3 theorems, 51 equations, 15 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Methodology
  3. Problem Definition
  4. Polynomial Stacking Neural Operator Framework
  5. Polynomial Layer: Chebyshev Basis and Modulated Transformation
  6. Chebyshev Spectral Basis
  7. Modulated Transformation
  8. Optimization Objectives and Loss Functions
  9. Theoretical Analysis of Chebyshev Polynomials Neural Operators
  10. Chebyshev Spectral Approximation and Error Decomposition
  11. Estimation Error Bounds and Numerical Stability
  12. Theoretical Analysis of Numerical Stability in Backward Propagation
  13. Experiment Results and Discussions
  14. Comparative Numerical Results
  15. Converge Speed and Accuracy Study
  16. ...and 10 more sections

Key Result

Theorem 1

Let $u(x;\theta)$ be analytic within a Bernstein ellipse $\mathcal{E}_{\rho}$ with foci at $\pm 1$ and semi-major axis summed with semi-minor axis equal to $\rho > 1$. The truncation error of the Chebyshev series satisfies: where $C$ is a constant dependent on the maximum modulus of $u$ within $\mathcal{E}_{\rho}$.

Figures (15)

  • Figure 1: Classification of PDE solving methods by their ability to handle single vs parametric PDEs and data requirements.
  • Figure 2: The main framework of the physics informed Chebyshev polynomial neural operator (CPNO)
  • Figure 3: Comparison of Convergence Curves for Different Methods on test cases.
  • Figure 4: Comparison of Minimum Training Epochs Required to Reach Specific Test Accuracies for Different Methods on test cases.
  • Figure 5: Visual Comparison of Predicted Solutions, Error Contours, and Error Spectrum Contours for Burgers.
  • ...and 10 more figures

Theorems & Definitions (3)

  • Theorem 1: Exponential Decay of Truncation Error
  • Theorem 2: Bound on Function Amplitude
  • Theorem 3: Bound on Derivative Amplitude