Data-driven discovery of partial differential equations (PDEs) has achieved considerable development in recent years. Several aspects of problems have been resolved by sparse regression-based and neural network-based methods. However, the performances of existing methods lack stability when dealing with complex situations, including sparse data with high noise, high-order derivatives and shock waves, which bring obstacles to calculating derivatives accurately. Therefore, a robust PDE discovery framework, called the robust deep learning-genetic algorithm (R-DLGA), that incorporates the physics-informed neural network (PINN), is proposed in this work. In the framework, a preliminary result of potential terms provided by the deep learning-genetic algorithm is added into the loss function of the PINN as physical constraints to improve the accuracy of derivative calculation. It assists to optimize the preliminary result and obtain the ultimately discovered PDE by eliminating the error compensation terms. The stability and accuracy of the proposed R-DLGA in several complex situations are examined for proof-and-concept, and the results prove that the proposed framework is able to calculate derivatives accurately with the optimization of PINN and possesses surprising robustness to complex situations, including sparse data with high noise, high-order derivatives, and shock waves.
翻译:近年来,数据驱动的局部差异方程式(PDEs)发现数据驱动部分差异方程式(PDEs)的发现取得了相当大的进展,这些问题的有些方面已经通过基于回归和神经网络的方法稀少解决,但是,在处理复杂情况时,现有方法的性能缺乏稳定性,包括高噪音、高序列衍生物衍生物和冲击波数据稀少,从而给准确计算衍生物带来障碍。因此,在这项工作中提出了强有力的PDE发现框架,称为强健的深学习-基因算法(R-DLGA),其中包括物理知情神经网络(PINN),在框架中,深学习-基因算法提供的潜在术语的初步结果被添加到PINN的损失功能中,作为提高衍生物计算准确性的物理限制。它协助优化初步结果,并通过消除错误补偿条件最终发现PDE。对若干复杂情况下拟议的R-DLGA(R-DLGA)的稳定性和准确性进行了检查,结果证明,拟议的框架能够准确计算出衍生物与PINN的优化,并具有惊人的稳健态,包括高噪音和高波震荡。