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<p>Solving partial differential equations with neural networks is an increasingly feasible approach, particularly for complex systems in fluid mechanics. Physics-Informed Neural Networks are a prominent data-free method where a loss function derived from the governing PDEs is minimized to approximate the solution. However, standard PINNs often face challenges such as solution instability. This research introduces an innovative PINN architecture that achieves higher accuracy in solving the Navier–Stokes equations by combining gating techniques, Fourier features, and trainable activation functions to effectively learn steep gradients. Furthermore, the model facilitates the enforcement of hard boundary conditions by replacing automatic differentiation with finite difference approximations, simplifying optimization and significantly reducing optimizer bias. The network's robustness is demonstrated on a suite of challenging benchmarks, including lid-driven cavity flow at high Reynolds numbers (up to 3200) and natural convection at a high Rayleigh number (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({10}^{7}\)</EquationSource> </InlineEquation>). In these regimes, the proposed model overcomes common instabilities and achieves competitive accuracy and computational efficiency compared to previous PINN-based methods.</p>

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A novel physics-informed neural network architecture for solving nonlinear PDEs in fluid mechanics without labeled data

  • 期刊论文

  • 发表日期：

    2026年2月17日

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  • Hossein Vasheghani Farahani,
  • Amir Mahdi Tahsini,
  • Ali Nematollahi,
  • Milad Ghiasi Afzal

摘要

Solving partial differential equations with neural networks is an increasingly feasible approach, particularly for complex systems in fluid mechanics. Physics-Informed Neural Networks are a prominent data-free method where a loss function derived from the governing PDEs is minimized to approximate the solution. However, standard PINNs often face challenges such as solution instability. This research introduces an innovative PINN architecture that achieves higher accuracy in solving the Navier–Stokes equations by combining gating techniques, Fourier features, and trainable activation functions to effectively learn steep gradients. Furthermore, the model facilitates the enforcement of hard boundary conditions by replacing automatic differentiation with finite difference approximations, simplifying optimization and significantly reducing optimizer bias. The network's robustness is demonstrated on a suite of challenging benchmarks, including lid-driven cavity flow at high Reynolds numbers (up to 3200) and natural convection at a high Rayleigh number ( \({10}^{7}\) ). In these regimes, the proposed model overcomes common instabilities and achieves competitive accuracy and computational efficiency compared to previous PINN-based methods.