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<p>The Physics-Informed Deep Operator Network (PI-DeepONet) is a class of neural networks that incorporates physics-based constraints through regularization mechanisms. However, its application to complex nonlinear parametric PDEs is often constrained by computational inefficiency and limited accuracy. In this paper, we propose a hybrid model that integrates the standard PI-DeepONet with a lightweight, single-layer artificial neural network (ANN) that employs Chebyshev (ChNN) or Legendre (LeNN) polynomial bases. The key idea behind this combination is to leverage the intrinsic benefits of these orthogonal polynomial networks–namely, their ability to handle sharp gradients via input expansion, their convergence and stability for capturing nonlinear behaviors, and their requirement for fewer trainable parameters–to complement and enhance the physics-informed learning of the DeepONet. Numerical results demonstrate that our proposed hybrid method achieves significant accuracy improvements compared to the standard PI-DeepONet, while maintaining computational efficiency without introducing significant extra cost. In fact, it reduces the number of training iterations required to achieve comparable accuracy, thereby improving overall computational efficiency. Extensive validations on a nonlinear parametric reaction-diffusion equation and two-dimensional (2D) Eikonal equations with circular and airfoil geometries confirm the superior performance of the proposed scheme in terms of accuracy, stability, generalization ability and training convergence.</p>

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A novel hybrid Physics-Informed DeepONet augmented by single-layer orthogonal chebyshev and legendre networks for nonlinear parametric partial differential equations

  • 期刊论文

  • 发表日期：

    2026年6月30日

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  • Mostafa Bayat,
  • Mehdi Dehghan,
  • Mostafa Abbaszadeh

摘要

The Physics-Informed Deep Operator Network (PI-DeepONet) is a class of neural networks that incorporates physics-based constraints through regularization mechanisms. However, its application to complex nonlinear parametric PDEs is often constrained by computational inefficiency and limited accuracy. In this paper, we propose a hybrid model that integrates the standard PI-DeepONet with a lightweight, single-layer artificial neural network (ANN) that employs Chebyshev (ChNN) or Legendre (LeNN) polynomial bases. The key idea behind this combination is to leverage the intrinsic benefits of these orthogonal polynomial networks–namely, their ability to handle sharp gradients via input expansion, their convergence and stability for capturing nonlinear behaviors, and their requirement for fewer trainable parameters–to complement and enhance the physics-informed learning of the DeepONet. Numerical results demonstrate that our proposed hybrid method achieves significant accuracy improvements compared to the standard PI-DeepONet, while maintaining computational efficiency without introducing significant extra cost. In fact, it reduces the number of training iterations required to achieve comparable accuracy, thereby improving overall computational efficiency. Extensive validations on a nonlinear parametric reaction-diffusion equation and two-dimensional (2D) Eikonal equations with circular and airfoil geometries confirm the superior performance of the proposed scheme in terms of accuracy, stability, generalization ability and training convergence.