Figure 12 ,Table 2
    • Figure 1.  The framework of T-PINN-LBM.

    • Figure 2.  Schematic diagram of square-cavity flow showing domain and boundary conditions.

    • Figure 3.  Mean absolute error for models using Xavier initialization and TRWI with different numbers of configuration points.

    • Figure 4.  At Re = 100: from left to right, these figures depict (a) the reference solution, the predicted solutions of (b) PINN, (c) PINN-LBM, and (d) T-PINN-LBM of the lid-driven cavity flow.

    • Figure 5.  At Re = 100: from left to right, these figures depict the absolute error between the predicted solutions of (a) PINN, (b) PINN-LBM, and (c) T-PINN-LBM and the reference solution of the lid-driven cavity flow.

    • Figure 6.  Comparison of the velocity slices of PINN, PINN-LBM and T-PINN-LBM with the reference solution at position (a) x = 0.5 and (b) y = 0.5.

    • Figure 7.  Visualization of the relative error comparison for the lid-driven cavity across different network structures and models.

    • Figure 8.  Solving for prediction errors in the lid-driven cavity flow with different Re.

    • Figure 9.  At Re = 500: comparison of predicted values of (a) the reference solution, (b) PINN, and (c) T-PINN-LBM.

    • Figure 10.  At Re = 1000: comparison of predicted values of (a) the reference solution, (b) PINN, and (c) T-PINN-LBM.

    • Figure 11.  At Re = 2500: comparison of predicted values of (a) the reference solution, (b) PINN, and (c) T-PINN-LBM.

    • Figure 12.  At Re = 4000: comparison of predicted values of (a) the reference solution, (b) PINN, and (c) T-PINN-LBM.