基于守恒量的两阶段物理信息神经网络方法及其在局域波解中的应用
A two-stage physics-informed neural network method based on conserved quantities and applications in localized wave solutions
摘要：
基于物理信息的神经网络方法具有计算速度快、精度高等优点，为非线性偏微分方程的数值求解开辟了一条新途径。基于守恒量，我们设计了一个两阶段的PINN方法，该方法通过将物理系统的特征引入神经网络来适应方程的性质。它的显著优势在于，它可以从全球角度施加实际限制。在第一阶段，应用原始PINN。在第二阶段中，我们另外将守恒量的测量引入到均方误差损失中来训练神经网络。这种两阶段PINN方法被用来模拟丰富的局域波解的可积方程。我们主要研究Sawada-Kotera方程以及耦合方程：经典Boussinesq-Burgers方程组，得到了数据驱动孤子分子、M型双峰孤子、平台孤子、相互作用解，数值结果表明，这些解的丰富的动力学行为可以很好地再现和两个-与原始PINN方法相比，分段PINN方法可以显著提高预测精度，增强泛化能力。
英文摘要：
With the advantages of fast calculating speed and high precision, the physics-informed neural network method opens up a new approach for numerically solving nonlinear partial differential equations. Based on conserved quantities, we devise a two-stage PINN method which is tailored to the nature of equations by introducing features of physical systems into neural networks. Its remarkable advantage lies in that it can impose physical constraints from a global perspective. In stage one, the original PINN is applied. In stage two, we additionally introduce the measurement of conserved quantities into mean squared error loss to train neural networks. This two-stage PINN method is utilized to simulate abundant localized wave solutions of integrable equations. We mainly study the Sawada-Kotera equation as well as the coupled equations: the classical Boussinesq-Burgers equations and acquire the data-driven soliton molecule, M-shape double-peak soliton, plateau soliton, interaction solution, etc. Numerical results illustrate that abundant dynamic behaviors of these solutions can be well reproduced and the two-stage PINN method can remarkably improve prediction accuracy and enhance the ability of generalization compared to the original PINN method.
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