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

arXiv: https://arxiv.org/abs/2602.01737 · HTML: https://mlsanity.com/paper/2602.01737/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 47 references that are in the map.
- First posted on arXiv 2 Feb 2026.

## Builds on (most cited first, 18 of 18)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [HyperPINN: Learning parameterized differential equations with physics-informed hypernetworks](https://mlsanity.com/paper/2111.01008/) (2021)
- [Meta-Auto-Decoder for Solving Parametric Partial Differential Equations](https://mlsanity.com/paper/2111.08823/) (2021)
- [Polynomial Implicit Neural Representations For Large Diverse Datasets](https://mlsanity.com/paper/2303.11424/) (2023)
- [Theory-guided Auto-Encoder for Surrogate Construction and Inverse Modeling](https://mlsanity.com/paper/2011.08618/) (2020)
- [Physics-informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media](https://mlsanity.com/paper/2212.08965/) (2022)
- [PINNsFailureRegion Localization and Refinement through White-box AdversarialAttack](https://mlsanity.com/paper/2310.11789/) (2023)

## Built on by (most cited first, 1 of 1)

- [Physics-Informed Method of Group Data Handling: Adaptive Construction of Functional Representations with an Application to the Navier-Stokes Equations](https://mlsanity.com/paper/2609.39291/) (2026)

## Abstract (arXiv)

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.