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<p>In this paper, the Physics-Informed Neural Networks (PINNs) framework is utilized with a locally adaptive activation mechanism to solve nonlinear dispersive wave equations, which we refer to in this study as “Enhanced PINNs.” The architecture of Enhanced PINNs incorporates a layer-wise locally adaptive activation function together with a slope recovery regularization term to improve gradient propagation and training stability. The adaptive activation introduces trainable slope parameters that dynamically adjust during optimization, while the slope recovery term accelerates their growth in the early stages of training, enabling the network to better capture complex nonlinear solution structures. Within this Enhanced PINNs, the governing partial differential equations (PDEs), along with the associated initial and boundary conditions, are enforced through a unified physics-informed loss formulation. The effectiveness of the Enhanced PINNs is demonstrated on three representative nonlinear dispersive models: the Benjamin–Bona–Mahony (BBM) equation, the Modified Benjamin–Bona–Mahony (MBBM) equation, and the Benjamin–Bona–Mahony–Burgers (BBM–Burgers) equation. Numerical experiments show that the Enhanced PINNs consistently improves solution accuracy, convergence behavior, and robustness compared with standard PINNs. In particular, the Enhanced PINNs achieves lower relative <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L_2\)</EquationSource> </InlineEquation> errors and reduced computational cost, demonstrating its effectiveness as a mesh-free and data-efficient approach for solving nonlinear dispersive and dispersive–dissipative PDEs.</p>

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An enhanced physics-informed neural network with adaptive activation function for nonlinear dispersive wave equations

  • 期刊论文

  • 发表日期：

    2026年6月30日

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  • Harender Kumar,
  • Neha Yadav,
  • Sandeep Kumar

摘要

In this paper, the Physics-Informed Neural Networks (PINNs) framework is utilized with a locally adaptive activation mechanism to solve nonlinear dispersive wave equations, which we refer to in this study as “Enhanced PINNs.” The architecture of Enhanced PINNs incorporates a layer-wise locally adaptive activation function together with a slope recovery regularization term to improve gradient propagation and training stability. The adaptive activation introduces trainable slope parameters that dynamically adjust during optimization, while the slope recovery term accelerates their growth in the early stages of training, enabling the network to better capture complex nonlinear solution structures. Within this Enhanced PINNs, the governing partial differential equations (PDEs), along with the associated initial and boundary conditions, are enforced through a unified physics-informed loss formulation. The effectiveness of the Enhanced PINNs is demonstrated on three representative nonlinear dispersive models: the Benjamin–Bona–Mahony (BBM) equation, the Modified Benjamin–Bona–Mahony (MBBM) equation, and the Benjamin–Bona–Mahony–Burgers (BBM–Burgers) equation. Numerical experiments show that the Enhanced PINNs consistently improves solution accuracy, convergence behavior, and robustness compared with standard PINNs. In particular, the Enhanced PINNs achieves lower relative \(L_2\) errors and reduced computational cost, demonstrating its effectiveness as a mesh-free and data-efficient approach for solving nonlinear dispersive and dispersive–dissipative PDEs.