Abstract

In this paper, physics-informed liquid networks (PILNs) are proposed based on liquid time-constant networks (LTC) for solving nonlinear partial differential equations (PDEs). In this approach, the network state is controlled via ordinary differential equations (ODEs). The significant advantage is that neurons controlled by ODEs are more expressive compared to simple activation functions. In addition, the PILNs use difference schemes instead of automatic differentiation to construct the residuals of PDEs, which avoid information loss in the neighborhood of sampling points. As this method draws on both the traveling wave method and physics-informed neural networks (PINNs), it has a better physical interpretation. Finally, the KdV equation and the nonlinear Schrödinger equation are solved to test the generalization ability of the PILNs. To the best of the authors’ knowledge, this is the first deep learning method that uses ODEs to simulate the numerical solutions of PDEs.

中文翻译：

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基于液体时间常数网络求解非线性偏微分方程的新方法

摘要

在本文中，基于液体时间常数网络（LTC）提出了物理信息液体网络（PILN）来求解非线性偏微分方程（PDE）。在这种方法中，网络状态通过常微分方程 (ODE) 进行控制。显着的优点是，与简单的激活函数相比，ODE 控制的神经元更具表达能力。此外，PILN使用差分方案而不是自动微分来构造偏微分方程的残差，这避免了采样点邻域的信息丢失。由于该方法同时利用了行波法和物理信息神经网络（PINN），因此具有更好的物理解释。最后，求解KdV方程和非线性薛定谔方程来测试PILN的泛化能力。据作者所知，这是第一个使用 ODE 来模拟 PDE 数值解的深度学习方法。