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

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.

💡 Research Summary

The paper introduces the Physics‑Informed Chebyshev Polynomial Neural Operator (CPNO), a mesh‑free neural operator designed to overcome the spectral bias and numerical instability that plague existing multilayer perceptron (MLP)‑based operators when trained solely with physics‑informed losses. The authors observe that MLPs, with fixed pointwise nonlinearities, preferentially capture low‑frequency components and struggle with the high‑frequency, multi‑scale features typical of parametric PDE solutions. To address this, CPNO replaces the monomial basis traditionally implicit in MLPs with a Chebyshev spectral basis, which is the unique minimax polynomial family on the interval (