A physics informed neural network (PINN) incorporates the physics of a system by satisfying its boundary value problem through a neural network's loss function. The PINN approach has shown great success in approximating the map between the solution of a partial differential equation (PDE) and its spatio-temporal input. However, for strongly non-linear and higher order partial differential equations PINN's accuracy reduces significantly. To resolve this problem, we propose a novel PINN scheme that solves the PDE sequentially over successive time segments using a single neural network. The key idea is to re-train the same neural network for solving the PDE over successive time segments while satisfying the already obtained solution for all previous time segments. Thus it is named as backward compatible PINN (bc-PINN). To illustrate the advantages of bc-PINN, we have used the Cahn Hilliard and Allen Cahn equations, which are widely used to describe phase separation and reaction diffusion systems. Our results show significant improvement in accuracy over the PINN method while using a smaller number of collocation points. Additionally, we have shown that using the phase space technique for a higher order PDE could further improve the accuracy and efficiency of the bc-PINN scheme.

中文翻译：

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用于瞬态非线性和高阶偏微分方程的物理信息神经网络

物理信息神经网络 (PINN) 通过神经网络的损失函数满足系统的边界值问题来整合系统的物理特性。PINN 方法在逼近偏微分方程 (PDE) 的解与其时空输入之间的映射方面取得了巨大成功。然而，对于强非线性和高阶偏微分方程，PINN 的精度显着降低。为了解决这个问题，我们提出了一种新颖的 PINN 方案，该方案使用单个神经网络在连续的时间段上按顺序求解 PDE。关键思想是重新训练相同的神经网络来解决连续时间段的 PDE，同时满足所有先前时间段的已经获得的解决方案。因此它被命名为向后兼容的 PINN (bc-PINN)。为了说明 bc-PINN 的优势，我们使用了 Cahn Hilliard 和 Allen Cahn 方程，它们被广泛用于描述相分离和反应扩散系统。我们的结果表明，在使用较少数量的搭配点的情况下，与 PINN 方法相比，准确性有了显着提高。此外，我们已经表明，将相空间技术用于更高阶的 PDE 可以进一步提高 bc-PINN 方案的准确性和效率。