Novel Soliton Solutions to KdV-Type Equations Based on Physics-Informed Neural Networks
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摘要: 该文采用物理信息神经网络(physics-informed neural network，PINN)方法结合广义Miura变换，深入研究了三个KdV类方程，获得了一系列新的孤子解. 具体而言，研究成果包括：基于改进的PINN方法，获得了mKdV方程的扭结-钟形解的解析形式；通过Miura变换，发现了KdV方程的新单孤子解；结合广义Miura变换与PINN方法，预测出非线性较强的KdV类方程的暗孤子解. 通过将PINN方法的数值结果与理论分析结果进行对比可以得知，基于广义Miura变换的PINN方法是发现偏微分方程新数值解的有效途径，同时对理论研究具有重要的启示意义.Abstract: Physics-informed neural networks (PINNs) were applied in combination with generalized Miura transformations to investigate 3 KdV-type equations. Several novel soliton solutions, including the kink-bell solution of the mKdV equation, were derived analytically with the improved PINN method; a single-soliton-like solution of the KdV equation, was achieved through the Miura transformation; and a dark-soliton solution of a strongly nonlinear KdV equation, was obtained by means of both the generalized Miura transformation and the PINN methods. Comparison of the numerical results obtained under the PINN framework with the exact solutions from theoretical analysis shows that, the proposed algorithm effectively uncovers new numerical solutions of partial differential equations and offers valuable insights for theoretical research.
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图 1 mKdV方程的扭结-钟形解(23)
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 1. Kink-bell solution (23) to the mKdV equation
图 3 KdV方程的类单孤子解与单孤子解的对比
Figure 3. The comparison between the single-soliton-like solution and the single-soliton solution of the KdV equation
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