Physics-Informed Chebyshev Neural Operator

• CPNO is a mesh-free operator learning framework that leverages Chebyshev polynomial bases for enhanced stability and spectral accuracy in solving parameterized PDEs.
• It incorporates a parameter-dependent modulation mechanism within a deep neural network to effectively integrate heterogeneous, multi-scale, and high-frequency PDE data.
• Empirical results on benchmarks demonstrate state-of-the-art performance with rapid convergence and reduced errors in complex scenarios such as transonic airfoil flows.

The Physics-Informed Chebyshev Polynomial Neural Operator (CPNO) is a mesh-free deep learning framework devised to solve parameterized partial differential equations (PDEs). It replaces monomial or Fourier-type feature expansions with Chebyshev polynomial bases, thereby ensuring enhanced numerical stability, spectral convergence, and robustness in physics-informed learning settings. CPNO introduces a parameter-dependent modulation mechanism to seamlessly integrate heterogeneous PDE data, making it particularly effective for multi-scale, high-frequency, and parametric PDE operator learning, including applications in complex geometries such as transonic airfoil flow (Chen et al., 2 Feb 2026).

1. Chebyshev Polynomial Expansions and Functional Representation

CPNO employs the Chebyshev polynomials of the first kind, $T_n(x)$, for stable and efficient representation of functions defined on $[-1,1]$. Formally,

$$T_0(x)=1,\quad T_1(x)=x,\quad T_{n+1}(x)=2\,x\,T_n(x) - T_{n-1}(x),\quad x\in[-1,1].$$

These polynomials are orthogonal under the weight $(1-x^2)^{-1/2}$, and satisfy a uniform bound $|T_n(x)|\le 1$, making them suitable for constructing spectral approximations. For a sufficiently smooth function $u(x)$,

$u(x)\approx \sum_{k=0}^N c_k\,T_k(x),$

with coefficients $c_k$ given by a weighted inner product. For multi-dimensional domains (including time and parameters), tensor-product bases are used, ensuring that feature representations encompass each spatial/temporal and parametric input (Chen et al., 2 Feb 2026, Mostajeran et al., 6 Jan 2025).

2. CPNO Architectural Design

CPNO’s core is a $q$-layer synthesis network that recursively builds polynomial-valued feature maps. The initial feature vector $\mathbf h^{(0)}(x, t)$ is a concatenation of Chebyshev features evaluated coordinate-wise (inputs normalized to $[-1, 1]$). Each subsequent layer performs

$$\mathbf h^{(j)}(x, t; \theta) = \sigma\Big[(A_j\,\Phi(x, t))\,\odot\,\mathbf h^{(j-1)}(x, t)\odot\boldsymbol{\omega}_j(\theta) + \boldsymbol{\varphi}_j(\theta)\Big],$$

where $A_j$ is a learnable linear spectral projection, $\odot$ is Hadamard (elementwise) product (effectively raising polynomial degree), and $$\boldsymbol\omega_j(\theta), \boldsymbol\varphi_j(\theta)$$ are parametric modulation vectors output by mapping networks ingesting PDE parameters $\theta$. The nonlinearity $\sigma(\cdot)$ is set to GELU or tanh to ensure smoothness and spectral fidelity. After $q$ layers, a final affine map outputs the PDE solution or its expansion coefficients.

Pseudocode summarizing the network forward pass:

x_, t_ = normalize(x, t) # to [-1, 1] Φ = ChebyshevFeatures(x_, t_) h = Φ for j in range(q): p = A_j @ Φ s = p * h ω, φ = MappingNet_j(θ) h = GELU(ω * s + φ) u_pred = W_q @ h + b_q return u_pred

Each mapping network $M_j$ is a small MLP (2 layers, width 32 in experiments), ingesting finitely-encoded PDE parameters (e.g., Chebyshev coefficients of coefficient functions or scalars like viscosity) (Chen et al., 2 Feb 2026).

3. Physics-Informed Loss Formulation

The network is trained by minimizing a composite loss: $$\mathcal L(\phi) = \lambda_{\rm pde}\,\mathcal L_{\rm pde} + \lambda_{\rm ic}\,\mathcal L_{\rm ic} + \lambda_{\rm bc}\,\mathcal L_{\rm bc} + \lambda_{\rm data}\,\mathcal L_{\rm data}.$$

• PDE Residual: $$\mathcal L_{\rm pde} = \mathbb E_{x, t, \theta} \|\mathcal L(\theta) u_\phi(x, t; \theta) - f(x, t; \theta)\|_2^2$$
• Initial Condition: $$\mathcal L_{\rm ic} = \mathbb E_{x, \theta} \|\mathcal I(\theta) u_\phi(0, x; \theta) - u_0(x; \theta)\|_2^2$$
• Boundary Condition: $$\mathcal L_{\rm bc} = \mathbb E_{x\in\partial\Omega, t, \theta} \|\mathcal B(\theta) u_\phi(x, t; \theta) - g(x, t; \theta)\|_2^2$$
• Data Fit (optional): $$\mathcal L_{\rm data} = \mathbb E_{(x, t, \theta, u^{\rm ref})} \|u_\phi(x, t; \theta) - u^{\rm ref}\|_2^2$$

Hyperparameters $$\lambda_{\rm pde}, \lambda_{\rm ic}, \lambda_{\rm bc}, \lambda_{\rm data}$$ balance the loss components; typical values set PDE, initial, and boundary terms to $1.0$, data term (when used) to $10^{-3}$ (Chen et al., 2 Feb 2026).

4. Theoretical Properties and Numerical Conditioning

CPNO’s use of the Chebyshev basis addresses two critical operator learning challenges:

1. Spectral Bias Mitigation: The Chebyshev expansion exhibits near-minimax uniform approximation error. For $u(x; \theta)$ analytic on a Bernstein ellipse, the best $N$-term Chebyshev truncation satisfies

$$\|u(\cdot;\theta) - \sum_{k=0}^N c_k^*(\theta)T_k\|_{L^\infty([-1,1])} \leq \frac{C\rho^{-N}}{\rho - 1}$$

for some $\rho > 1$ and constant $C$.

2. Stability and Conditioning: The Lebesgue constant for Chebyshev nodes grows only logarithmically,

$\Lambda_N = \frac{2}{\pi}\ln(N+1) + O(1)$

and the condition number of the Chebyshev-gram matrix $O(N^2)$, dramatically lower than $O(e^{cN})$ for monomials. This ensures stable gradient propagation and robust training even at high polynomial orders (Chen et al., 2 Feb 2026, Mostajeran et al., 6 Jan 2025).

5. Benchmark Performance and Empirical Results

CPNO achieves state-of-the-art accuracy and convergence rates on a wide range of parameterized PDE benchmarks:

• Burgers’ equation: $L^2$ error $3.06\times10^{-2}$ (zero-shot, no data), outperforming PI-DeepONet ($8.09\times10^{-2}$) and HyperPINNs ($4.55\times10^{-2}$).
• Allen–Cahn equation: $1.69\times10^{-2}$ (zero-shot).
• Diffusion–Reaction: $2.96\times10^{-2}$ (zero-shot).
• 2D vorticity–Navier–Stokes: $2.72\times10^{-2}$ (zero-shot).
• Few-shot regime: Additional three solution snapshots reduce errors further (e.g., $1.09\times10^{-2}$ for Allen–Cahn).

The framework demonstrates rapid convergence: errors $<10^{-2}$ are typically attained in less than 5000 epochs, whereas baseline neural operator models require over 20,000 epochs. In frequency analysis, CPNO captures high-wavenumber content ($k>20$) not resolved by MLP-based alternatives.

In the transonic airfoil flow experiment (parameterized complex geometry), CPNO with Chebyshev order 10, network depth 12, and degree-16 encoding achieves L1 errors on $(u, v, p)$ of mean $\sim 2\times10^{-3}$ and max $\sim 8\times10^{-2}$, demonstrating its capacity for operator learning on challenging fluid dynamics problems (Chen et al., 2 Feb 2026).

6. Implementation Best Practices

Recommended configuration for CPNO includes:

• Chebyshev polynomial order $P=4\sim 6$ for smooth fields; $P\leq 10$ for sharp layers or shocks.
• Depth $q=6-10$ layers; hidden dimension $d_h=(d+1)(P+1)$ (typically $64–128$).
• Activation: GELU or tanh; ReLU is discouraged due to nondifferentiability affecting spectral accuracy.
• Adam optimizer with learning rate warmed up to $10^{-3}$ and multiplicative decay.
• Minibatch of $2000–4000$ uniformly-sampled collocation points per step.
• Normalization of each physical domain to $[-1,1]$ to ensure polynomial features remain well-scaled.
• Mapping networks for parameter-dependent modulation are kept shallow and narrow (2–3 layers, width 16–32) to control overfitting (Chen et al., 2 Feb 2026).

7. Extensions, Limitations, and Context

CPNO’s foundation explicitly incorporates the Chebyshev basis, parameter modulation, and mesh-free operator learning, distinguishing it from monomial-MLPs, Fourier-based, or vanilla Galerkin and KAN variants (Guo et al., 2024, Zhang et al., 13 May 2025, Mostajeran et al., 6 Jan 2025). While CPNO is robust to multi-scale and high-frequency phenomena, practical limitations include the expense of high-dimensional tensor-product expansions and domain normalization strategies for non-rectilinear geometries. Potential future work involves sparse-grid Chebyshev representations, adaptive network depth, and hybridization with attention-based enhancements as in recent Chebyshev–KAN models (Zhang et al., 13 May 2025).

CPNO situates itself within the rapidly evolving landscape of neural operator design, providing a theoretically principled and empirically validated approach to solving parametric, time-dependent, and nonlinear PDEs with superior accuracy, efficiency, and training robustness (Chen et al., 2 Feb 2026).