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提出了扩展混合训练物理信息神经网络(X-MTPINNs), 该模型通过整合扩展物理信息神经网络(X-PINNs)的域分解技术与混合训练物理信息神经网络(MTPINNs)框架, 有效提升了非线性波动问题的求解能力. 相较于经典物理信息神经网络(PINNs)模型, 新模型具有双重优势: 1)混合训练框架通过优化初边值条件的处理机制, 显著改善了模型收敛特性, 在提升非线性波解拟合精度的同时, 将计算时间降低约40%; 2) X-PINNs的域分解技术增强了模型对复杂动力学行为的表征能力. 基于非线性薛定谔方程(NLSE)的数值实验表明, X-MTPINNs在亮双孤子解及三阶怪波求解以及参数反演等任务中均表现优异, 其预测精度较传统PINNs提升一至两个数量级. 对于逆问题, X-MTPINNs算法在有噪声和无噪声条件下都能准确识别NLSE中的未知参数, 解决了经典PINNs在本研究条件下NLSE参数识别中完全失效的问题, 表现出很强的鲁棒性.
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关键词:
- 物理信息神经网络 /
- 非线性薛定谔方程 /
- 扩展混合训练物理信息神经网络 /
- 域分解 /
- 参数发现
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.-
Keywords:
- physics-informed neural networks /
- nonlinear Schrödinger equation /
- extended mixed-training physics-informed neural networks /
- domain decomposition /
- parameter identification
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] -
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 4.52×10–2 2531.88 [–3.30, –2.50] 4 50 2.63×10–2 2459.23 [–2.50, –1.40] 4 50 3.03×10–2 2347.56 [–1.40, 2.00] 4 50 5.73×10–2 2489.87 [2.00, 5.00] 4 50 1.70×10–2 2489.87 Global [–5, 5] 4 50 3.38×10–2 2531.88 PINNs [–5, 5] 4 50 2.51×10–1 4176.69 
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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 1.25×10–3 3065.07 [–0.60, –0.15] 4 50 1.22×10–3 2228.26 [–0.15, 0.15] 4 50 3.89×10–3 2018.29 [0.15, 0.60] 4 50 1.31×10–3 2438.84 [0.60, 2.00] 4 50 1.84×10–3 2339.57 Global [–2, 2] 4 50 2.25×10–3 3065.07 PINNs [–2, 2] 4 50 4.94×10–1 5538.17
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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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[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
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