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A parameter-driven physics-informed neural network framework for solving two-parameter singular perturbation problems involving boundary layers

  • *Corresponding author: Srinivasan Natesan

    *Corresponding author: Srinivasan Natesan
Abstract / Introduction Full Text(HTML) Figure(14) / Table(15) Related Papers Cited by
  • In this article, our goal is to solve two-parameter singular perturbation problems (SPPs) in one- and two-dimensions using an adapted Physics-Informed Neural Networks (PINNs) approach. Such problems are of major importance in engineering and the sciences, as they arise in control theory, fluid and gas dynamics, financial modeling, and related areas. These problems often exhibit boundary and/or interior layers, making them particularly challenging to solve. Prior studies have shown that standard PINNs suffer from low accuracy and struggle to effectively address such issues. A recently proposed enhancement, known as the parameter asymptotic PINNs (PA-PINNs), has demonstrated improved performance over standard PINNs and Gradient enhanced PINNs (gPINNs) in handling one-parameter singularly perturbed convection-dominated problems, offering better accuracy, convergence, and stability. In this article, we extend and evaluate the robustness of PA-PINNs for the first time in solving two-parameter SPPs. Furthermore, we derive bounds on the generalization error, linking training residuals to quadrature errors, thereby providing a theoretical foundation for the reliability of PINNs in solving multi-scale two-parameter SPPs. We also present a comprehensive comparison with standard finite difference and finite element methods, analyzing accuracy and computational efficiency across various neural network architectures.

    Mathematics Subject Classification: 65M12, 68T07, 35J30, 76M45.

    Citation:

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  • Figure 1.  A basic feed-forward neural network

    Figure 2.  PA-PINNs Algorithm flowchart

    Figure 3.  Comparison between Exact and PA-PINNs solution for different values of perturbation parameter $ \varepsilon_1, \varepsilon_2 $ for Example 5.1

    Figure 4.  Comparison between Exact and PINNs solution for different values of perturbation parameter $ \varepsilon_1, \varepsilon_2 $ for Example 5.2

    Figure 5.  Comparison between Exact and PA-PINNs solution for different values of perturbation parameter $ \varepsilon_1, \varepsilon_2 $ for Example 5.4

    Figure 6.  Comparison between Exact and PA-PINNs solution for different values of perturbation parameter $ \varepsilon_1, \varepsilon_2 $ for Example 5.5

    Figure 7.  Error vs epoch plots for Example 5.1 with $ (\varepsilon_1, \varepsilon_2) = (10^{-3}, 10^{-4}) $

    Figure 8.  Loss curve for Example 5.1 with $ (\varepsilon_1, \varepsilon_2) = (10^{-3}, 10^{-4}) $

    Figure 9.  Error vs epoch plots for Example 5.2 with $ (\varepsilon_1, \varepsilon_2) = (10^{-3}, 10^{-4}) $

    Figure 10.  Loss curve for Example 5.2 with $ (\varepsilon_1, \varepsilon_2) = (10^{-3}, 10^{-4}) $

    Figure 11.  Error vs epoch plots for Example 5.4 with $ (\varepsilon_1, \varepsilon_2) = (10^{-3}, 10^{-4}) $

    Figure 12.  Loss curve for Example 5.4 with $ (\varepsilon_1, \varepsilon_2) = (10^{-3}, 10^{-4}) $

    Figure 13.  Error vs epoch plots for Example 5.6 with $ (\varepsilon_1, \varepsilon_2) = (10^{-3}, 10^{-4}) $

    Figure 14.  Loss curve for Example 5.6 with $ (\varepsilon_1, \varepsilon_2) = (10^{-3}, 10^{-4}) $

    Table 1.  Error comparison for various parameter values for Example 5.1

    Method $ \varepsilon_{1} $ $ \varepsilon_2 $ $ \mathscr{E}_{2} $
    PINN $ 10^{-2} $ $ 10^{-3} $ 1.661e-03
    $ 10^{-3} $ $ 10^{-4} $ 1.287e-02
    $ 10^{-4} $ $ 10^{-5} $ 7.905e-02
    $ 10^{-5} $ $ 10^{-6} $ 1.357e-01
    PA-PINN $ 10^{-2} $ $ 10^{-3} $ 2.446e-05
    $ 10^{-3} $ $ 10^{-4} $ 1.548e-04
    $ 10^{-4} $ $ 10^{-5} $ 1.782e-03
    $ 10^{-5} $ $ 10^{-6} $ 1.273e-02
     | Show Table
    DownLoad: CSV

    Table 2.  Error comparison for various parameter values for Example 5.2

    Method $ \varepsilon_{1} $ $ \varepsilon_2 $ $ \mathscr{E}_{2} $
    PINN $ 10^{-1} $ $ 10^{-2} $ 1.911e-04
    $ 10^{-2} $ $ 10^{-3} $ 2.312e-03
    $ 10^{-3} $ $ 10^{-4} $ 1.790e-01
    $ 10^{-4} $ $ 10^{-5} $ 4.577e-02
    PA-PINN $ 10^{-1} $ $ 10^{-2} $ 4.378e-04
    $ 10^{-2} $ $ 10^{-3} $ 2.859e-06
    $ 10^{-3} $ $ 10^{-4} $ 2.295e-05
    $ 10^{-4} $ $ 10^{-5} $ 8.580e-03
     | Show Table
    DownLoad: CSV

    Table 3.  Error comparison for various parameter values for Example 5.3

    Method $ \varepsilon_{1} $ $ \varepsilon_2 $ $ \mathscr{E}_{2} $
    PINN $ 10^{-1} $ $ 10^{-2} $ 8.023e-01
    $ 10^{-2} $ $ 10^{-3} $ 1.075e+00
    $ 10^{-3} $ $ 10^{-4} $ 8.255e-01
    $ 10^{-4} $ $ 10^{-5} $ 8.777e-01
    PA-PINN $ 10^{-1} $ $ 10^{-2} $ 1.897e-01
    $ 10^{-2} $ $ 10^{-3} $ 5.375e-01
    $ 10^{-3} $ $ 10^{-4} $ 6.269e-01
    $ 10^{-4} $ $ 10^{-5} $ 6.477e-01
     | Show Table
    DownLoad: CSV

    Table 4.  Error comparison for various parameter values for Example 5.4

    Method $ \varepsilon_{1} $ $ \varepsilon_2 $ $ \mathscr{E}_{2} $
    PINN $ 10^{-1} $ $ 10^{-2} $ 4.418e-02
    $ 10^{-2} $ $ 10^{-3} $ 8.363e-02
    $ 10^{-3} $ $ 10^{-4} $ 1.520e-01
    $ 10^{-4} $ $ 10^{-5} $ 2.467e-01
    PA-PINN $ 10^{-1} $ $ 10^{-2} $ 1.011e-04
    $ 10^{-2} $ $ 10^{-3} $ 1.084e-04
    $ 10^{-3} $ $ 10^{-4} $ 2.621e-04
    $ 10^{-4} $ $ 10^{-5} $ 3.812e-04
     | Show Table
    DownLoad: CSV

    Table 5.  Error comparison for various parameter values for Example 5.5

    Method $ \varepsilon_{1} $ $ \varepsilon_2 $ $ \mathscr{E}_{2} $
    PINN $ 10^{-1} $ $ 10^{-2} $ 4.680e-02
    $ 10^{-2} $ $ 10^{-3} $ 9.387e-02
    $ 10^{-3} $ $ 10^{-4} $ 2.352e-01
    $ 10^{-4} $ $ 10^{-5} $ 2.962e-01
    PA-PINN $ 10^{-1} $ $ 10^{-2} $ 5.609e-04
    $ 10^{-2} $ $ 10^{-3} $ 1.895e-04
    $ 10^{-3} $ $ 10^{-4} $ 2.232e-03
    $ 10^{-4} $ $ 10^{-5} $ 8.933e-03
     | Show Table
    DownLoad: CSV

    Table 6.  Error comparison for various parameter values for Example 5.6

    Method $ \varepsilon_{1} $ $ \varepsilon_2 $ $ \mathscr{E}_{2} $ $ \mathscr{E}_{\infty} $
    PINN $ 10^{-1} $ $ 10^{-2} $ 4.944e-01 2.999e-02
    $ 10^{-2} $ $ 10^{-3} $ 4.492e-01 6.970e-02
    $ 10^{-3} $ $ 10^{-4} $ 4.730e-01 9.653e-02
    $ 10^{-4} $ $ 10^{-5} $ 5.465e-01 1.192e-01
    PA-PINN $ 10^{-1} $ $ 10^{-2} $ 6.941e-04 2.363e-04
    $ 10^{-2} $ $ 10^{-3} $ 4.626e-04 2.918e-04
    $ 10^{-3} $ $ 10^{-4} $ 1.417e-03 1.130e-03
    $ 10^{-4} $ $ 10^{-5} $ 1.173e-03 4.361e-03
     | Show Table
    DownLoad: CSV

    Table 7.  Error comparison using nonlinear update with different widths and depths for Example 5.1

    Depth $\times$ Width $ (\varepsilon_1, \varepsilon_2) $ $ \mathscr{E}_2 $ CPU time
    $ 4 \times 20 $ $ (10^{-2}, 10^{-3}) $ 7.38608e-03 1379.94 sec
    $ (10^{-3}, 10^{-4}) $ 1.21742e-02 2137.08 sec
    $ 4 \times 40 $ $ (10^{-2}, 10^{-3}) $ 7.38585e-03 743.09 sec
    $ (10^{-3}, 10^{-4}) $ 1.21742e-02 1909.18 sec
    $ 8 \times 20 $ $ (10^{-2}, 10^{-3}) $ 7.38600e-03 1471.01 sec
    $ (10^{-3}, 10^{-4}) $ 1.21756e-02 3138.97 sec
    $ 8 \times 40 $ $ (10^{-2}, 10^{-3}) $ 7.38585e-03 1195.67 sec
    $ (10^{-3}, 10^{-4}) $ 1.21736e-02 3286.40 sec
     | Show Table
    DownLoad: CSV

    Table 8.  Error comparison using nonlinear update with different widths and depths for Example 5.2

    Depth $\times$ Width $ (\varepsilon_1, \varepsilon_2) $ $ \mathscr{E}_2 $ CPU time
    $ 4 \times 20 $ $ (10^{-2}, 10^{-3}) $ 3.67415e-03 997.53 sec
    $ (10^{-3}, 10^{-4}) $ 8.94421e-03 2055.41 sec
    $ 4 \times 40 $ $ (10^{-2}, 10^{-3}) $ 3.67407e-03 931.62 sec
    $ (10^{-3}, 10^{-4}) $ 8.94421e-03 1958.53 sec
    $ 8 \times 20 $ $ (10^{-2}, 10^{-3}) $ 3.67408e-03 1291.48 sec
    $ (10^{-3}, 10^{-4}) $ 8.94382e-03 3112.42 sec
    $ 8 \times 40 $ $ (10^{-2}, 10^{-3}) $ 3.67414e-03 1571.71 sec
    $ (10^{-3}, 10^{-4}) $ 8.94369e-03 3566.46 sec
     | Show Table
    DownLoad: CSV

    Table 9.  Error comparison using linear update with different widths and depths for Example 5.1

    Depth $\times$ Width $ (\varepsilon_1, \varepsilon_2) $ $ \mathscr{E}_2 $ CPU time
    $ 4 \times 20 $ $ (10^{-2}, 10^{-3}) $ 6.67240e-06 499.03 sec
    $ (10^{-3}, 10^{-4}) $ 5.08541e-04 673.43 sec
    $ 4 \times 40 $ $ (10^{-2}, 10^{-3}) $ 1.42943e-05 742.54 sec
    $ (10^{-3}, 10^{-4}) $ 2.47156e-04 929.41 sec
    $ 8 \times 20 $ $ (10^{-2}, 10^{-3}) $ 5.40406e-06 332.67 sec
    $ (10^{-3}, 10^{-4}) $ 4.68982e-04 966.12 sec
    $ 8 \times 40 $ $ (10^{-2}, 10^{-3}) $ 2.25027e-05 1115.13 sec
    $ (10^{-3}, 10^{-4}) $ 1.404431e-04 1615.46 sec
     | Show Table
    DownLoad: CSV

    Table 10.  Error comparison using linear update with different widths and depths for Example 5.2

    Depth $\times$ Width $ (\varepsilon_1, \varepsilon_2) $ $ \mathscr{E}_2 $ CPU time
    $ 4 \times 20 $ $ (10^{-2}, 10^{-3}) $ 3.14247e-06 365.56 sec
    $ (10^{-3}, 10^{-4}) $ 2.13913e-03 550.16 sec
    $ 4 \times 40 $ $ (10^{-2}, 10^{-3}) $ 8.42577e-06 346.05 sec
    $ (10^{-3}, 10^{-4}) $ 7.52966e-04 797.75 sec
    $ 8 \times 20 $ $ (10^{-2}, 10^{-3}) $ 4.24980e-06 482.25 sec
    $ (10^{-3}, 10^{-4}) $ 8.31980e-05 961.15 sec
    $ 8 \times 40 $ $ (10^{-2}, 10^{-3}) $ 3.18480e-06 615.23 sec
    $ (10^{-3}, 10^{-4}) $ 3.87194e-05 1405.22 sec
     | Show Table
    DownLoad: CSV

    Table 11.  Error comparison using linear update with different widths and depths for Example 5.3

    Depth $\times$ Width $ (\varepsilon_1, \varepsilon_2) $ $ \mathscr{E}_2 $ CPU time
    $ 2 \times 20 $ $ (10^{-2}, 10^{-3}) $ 5.19825e-01 442.16 sec
    $ (10^{-3}, 10^{-4}) $ 6.28389e-01 441.14 sec
    $ 4 \times 20 $ $ (10^{-2}, 10^{-3}) $ 5.39786e-01 648.12 sec
    $ (10^{-3}, 10^{-4}) $ 6.60805e-01 638.12 sec
    $ 4 \times 40 $ $ (10^{-2}, 10^{-3}) $ 5.39773e-01 807.41 sec
    $ (10^{-3}, 10^{-4}) $ 6.51049e-01 813.42 sec
    $ 8 \times 20 $ $ (10^{-2}, 10^{-3}) $ 5.39772e-01 1062.02 sec
    $ (10^{-3}, 10^{-4}) $ 6.52699e-01 1063.48 sec
    $ 8 \times 40 $ $ (10^{-2}, 10^{-3}) $ 5.39773e-01 1391.96 sec
    $ (10^{-3}, 10^{-4}) $ 6.28496e-01 1407.58 sec
     | Show Table
    DownLoad: CSV

    Table 12.  Error comparison using linear update with different widths and depths for Example 5.4

    Depth $\times$ Width $ (\varepsilon_1, \varepsilon_2) $ $ \mathscr{E}_2 $ CPU time
    $ 4 \times 20 $ $ (10^{-2}, 10^{-3}) $ 6.87200e-04 2756.71 sec
    $ (10^{-3}, 10^{-4}) $ 7.37420e-03 3608.83 sec
    $ 4 \times 40 $ $ (10^{-2}, 10^{-3}) $ 4.36345e-04 5491.24 sec
    $ (10^{-3}, 10^{-4}) $ 6.08680e-03 5703.67 sec
    $ 8 \times 20 $ $ (10^{-2}, 10^{-3}) $ 1.94381e-04 4363.29 sec
    $ (10^{-3}, 10^{-4}) $ 1.59990e-03 6173.14 sec
    $ 8 \times 40 $ $ (10^{-2}, 10^{-3}) $ 9.16608e-05 7239.75 sec
    $ (10^{-3}, 10^{-4}) $ 1.52576e-03 8390.88 sec
     | Show Table
    DownLoad: CSV

    Table 13.  Error comparison for Example 5.1

    Method $ (\varepsilon_1, \varepsilon_2) $ $ \mathscr{E}_{\infty} $
    Upwind FDM (over Shishkin mesh) with $ \mathcal{N} $ = 1024 $ (10^{-2}, 10^{-3}) $ 1.18356e-05
    $ (10^{-3}, 10^{-4}) $ 7.68553e-03
    $ (10^{-4}, 10^{-5}) $ 6.62190e-02
    Proposed method with depth $ \times $ width $ = 8 \times 20 $ and $ \eta_{o} = 1000 $ $ (10^{-2}, 10^{-3}) $ 1.26608e-05
    $ (10^{-3}, 10^{-4}) $ 9.75883e-04
    $ (10^{-4}, 10^{-5}) $ 4.50602e-03
     | Show Table
    DownLoad: CSV

    Table 14.  Error comparison for Example 5.2

    Method $ (\varepsilon_1, \varepsilon_2) $ $ \mathscr{E}_{\infty} $
    Upwind FDM (over Shishkin mesh) with $ \mathcal{N} $ = 1024 $ (10^{-1}, 10^{-2}) $ 1.13160e-02
    $ (10^{-2}, 10^{-3}) $ 7.27137e-03
    $ (10^{-3}, 10^{-4}) $ 7.72630e-03
    Standard FEM (over Shishkin mesh) with $ \mathcal{N} $ = 1024 $ (10^{-1}, 10^{-2}) $ 1.13324e-02
    $ (10^{-2}, 10^{-3}) $ 7.24019e-03
    $ (10^{-3}, 10^{-4}) $ 2.81312e-03
    Proposed method with depth $ \times $ width $ = 8 \times 20 $ and $ \eta_{o} = 1000 $ $ (10^{-1}, 10^{-2}) $ 5.08473e-03
    $ (10^{-2}, 10^{-3}) $ 9.27661e-06
    $ (10^{-3}, 10^{-4}) $ 6.79974e-04
     | Show Table
    DownLoad: CSV

    Table 15.  Error comparison for Example 5.4

    Method $ (\varepsilon_1, \varepsilon_2) $ $ \mathscr{E}_{\infty} $
    Standard FEM (over Shishkin mesh) with $ \mathcal{N} = 64 \times 64 \times 2 $ $ (10^{-2}, 10^{-3}) $ 6.68075e-04
    $ (10^{-3}, 10^{-4}) $ 6.87868e-03
    Proposed method with depth $ \times $ width $ = 8 \times 20 $ and $ \eta_{o} = 10k $ $ (10^{-2}, 10^{-3}) $ 1.94381e-04
    $ (10^{-3}, 10^{-4}) $ 1.59990e-03
     | Show Table
    DownLoad: CSV
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