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In recent years, physics-informed neural networks (PINNs) have provided effcient data-driven methods for solving forward and inverse problems of partial differential equations (PDEs). However, when addressing complex PDEs, PINNs face significant challenges in computational efficiency and accuracy. In this study, we propose the extended mixed-training physics-informed neural networks (X-MTPINNs) as illustrated in the following figure, which effectively enhance the ability to solve nonlinear wave problems by integrating the domain decomposition technique of extended physics-informed neural networks (X-PINNs) in a mixed-training physics-informed neural networks (MTPINNs) framework. Compared with the classical PINNs model, the new model exhibits dual advantages: The first advantage is that the mixed-training framework significantly improves convergence properties by optimizing the handling mechanism of initial and boundary conditions, achieving higher fitting accuracy for nonlinear wave solutions while reducing the computation time by approximately 40%. And the second advantage is that the domain decomposition technique from X-PINNs strengthens the ability of the model to represent complex dynamical behaviors. Numerical experiments based on the nonlinear Schrödinger equation (NLSE) demonstrate that X-MTPINNs excel perform well in solving two bright solitons, third-order rogue waves, and parameter inversion tasks, with prediction accuracy improved by one to two orders of magnitude over traditional PINN. For inverse problems, the X-MTPINNs algorithm accurately identifies unknown parameters in the NLSE under noise-free, 2%, and 5% noisy conditions, solving the complete failure problem of NSLE parameter identification in classical PINNs in the studied scenario, thus demonstrating strong robustness.
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Keywords:
- PINNs /
- NLSE /
- X-MTPINNs /
- domain decomposition /
- parameter identification
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图 1 区域分解示意图
Figure 1. Domain decomposition schematic diagram.
图 2 扩展混合训练物理信息神经网络(X-MTPINNs)
Figure 2. Extended Mixed-Training Physics-Informed Neural Networks (X-MTPINNs).
图 3 非线性薛定谔方程亮双孤子解的真实值和不同模型的预测误差
Figure 3. Exact values of two bright solitons in the nonlinear schrödinger equation and prediction errors from different models.
图 4 非线性薛定谔方程三阶怪波的真实值和不同模型的预测误差
Figure 4. Exact values of third-order rogue waves in the nonlinear schrödinger equation and prediction errors from different models.
表 1 两种模型对非线性薛定谔方程亮双孤子解的数值预测结果
Table 1. Numerical Prediction Results of Two Models for Two Bright Solitons in the Nonlinear Schrödinger Equation
亮双孤子 子域 隐藏
层数神经
元数L2误差 训练时间(s) X-MTPINNs [–5.00, –3.30] 4 50 4.52e-02 2531.88 [–3.30, –2.50] 4 50 2.63e-02 2459.23 [–2.50, –1.40] 4 50 3.03e-02 2347.56 [–1.40, 2.00] 4 50 5.73e-02 2489.87 [2.00, 5.00] 4 50 1.70e-02 2489.87 Global [–5, 5] 4 50 3.38e-02 2531.88 PINNs [–5, 5] 4 50 2.51e-01 4176.69 
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表 2 两种模型对非线性薛定谔方程三阶怪波的数值预测结果
Table 2. Numerical Prediction Results of Third-order Rogue Waves in the Nonlinear Schrödinger Equation
三阶怪波 子域 隐藏
层数神经
元数L2误差 训练时间(s) X-MTPINNs [–2.00, –0.60] 4 50 1.25e-03 3065.07 [–0.60, –0.15] 4 50 1.22e-03 2228.26 [–0.15, 0.15] 4 50 3.89e-03 2018.29 [0.15, 0.60] 4 50 1.31e-03 2438.84 [0.60, 2.00] 4 50 1.84e-03 2339.57 Global [–2, 2] 4 50 2.25e-03 3065.07 PINNs [–2, 2] 4 50 4.94e-01 5538.17
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表 3 通过学习所得的$ \lambda_1 $和$ \lambda_2 $及其误差
Table 3. Learned $ \lambda_1 $ and $ \lambda_2 $ and Their Associated Errors
NPDEs 噪声 非线性演化方程 相对误差$ [\lambda_1, \lambda_2] $(%) Correct – $ {\rm{i}}\hat{h}_t + 0.5\hat{h}_{xx} + |\hat{h}|^2\hat{h} = 0 $ $ [0, 0] $ [–2.00, –0.60] 0% $ \lambda_1 = 0.4999224, \lambda_2 = 0.9999491 $ $ [0.01553, 0.00509] $ 2% $ \lambda_1 = 0.5006779, \lambda_2 = 1.0010909 $ $ [0.13558, 0.10909] $ 5% $ \lambda_1 = 0.5017547, \lambda_2 = 1.0026274 $ $ [0.35094, 0.26274] $ [–0.60, –0.08] 0% $ \lambda_1 = 0.4995355, \lambda_2 = 0.9997470 $ $ [0.09289, 0.02530] $ 2% $ \lambda_1 = 0.4990919, \lambda_2 = 0.9979020 $ $ [0.18162, 0.20980] $ 5% $ \lambda_1 = 0.4983062, \lambda_2 = 0.9952767 $ $ [0.33876, 0.47233] $ [–0.08, 0.00] 0% $ \lambda_1 = 0.4907891, \lambda_2 = 0.9970491 $ $ [1.84218, 0.29509] $ 2% $ \lambda_1 = 0.4856022, \lambda_2 = 0.9931418 $ $ [2.87955, 0.68582] $ 5% $ \lambda_1 = 0.4838027, \lambda_2 = 0.9913622 $ $ [3.23946, 0.86378] $ [0.00, 0.08] 0% $ \lambda_1 = 0.4978686, \lambda_2 = 0.9941696 $ $ [0.42627, 0.58304] $ 2% $ \lambda_1 = 0.4953954, \lambda_2 = 0.9936743 $ $ [0.92092, 0.63257] $ 5% $ \lambda_1 = 0.4909737, \lambda_2 = 0.9887489 $ $ [1.80527, 1.12511] $ [0.08, 0.60] 0% $ \lambda_1 = 0.4999007, \lambda_2 = 0.9997627 $ $ [0.01987, 0.02373] $ 2% $ \lambda_1 = 0.4997132, \lambda_2 = 0.9992683 $ $ [0.05737, 0.07317] $ 5% $ \lambda_1 = 0.4974256, \lambda_2 = 0.9973150 $ $ [0.51489, 0.26850] $ [0.60, 2.00] 0% $ \lambda_1 = 0.5002598, \lambda_2 = 1.0001861 $ $ [0.05195, 0.01861] $ 2% $ \lambda_1 = 0.4997032, \lambda_2 = 0.9999705 $ $ [0.05936, 0.00295] $ 5% $ \lambda_1 = 0.4995049, \lambda_2 = 0.9998789 $ $ [0.09903, 0.01211] $ Global 0% $ \lambda_1 = 0.972926, \lambda_2 = 0.9987469 $ $ [0.54148, 0.12531] $ 2% $ \lambda_1 = 0.5035173, \lambda_2 = 0.9971460 $ $ [0.70345, 0.28540] $ 5% $ \lambda_1 = 4.9470665, \lambda_2 = 1.0050076 $ $ [1.05867, 0.50076] $
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