A New Method for Solving Nonlinear Partial Differential Equations Based on Liquid Time-Constant Networks

IF 2.6 3区 数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS
Jiuyun Sun, Huanhe Dong, Yong Fang
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引用次数: 0

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.

基于液体时常网络的非线性偏微分方程求解新方法
摘要 本文提出了基于液体时间不变网络(LTC)的物理信息液体网络(PILN),用于求解非线性偏微分方程(PDE)。在这种方法中,网络状态是通过常微分方程(ODE)控制的。其显著优势在于,与简单的激活函数相比,由 ODE 控制的神经元更具表现力。此外,PILNs 使用差分方案而不是自动微分来构建 PDE 的残差,从而避免了采样点附近的信息丢失。由于这种方法借鉴了行波法和物理信息神经网络(PINNs),因此具有更好的物理解释能力。最后,我们求解了 KdV 方程和非线性薛定谔方程,以检验 PILN 的泛化能力。据作者所知,这是第一种利用 ODE 模拟 PDE 数值解的深度学习方法。
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来源期刊
Journal of Systems Science & Complexity
Journal of Systems Science & Complexity 数学-数学跨学科应用
CiteScore
3.80
自引率
9.50%
发文量
90
审稿时长
6-12 weeks
期刊介绍: The Journal of Systems Science and Complexity is dedicated to publishing high quality papers on mathematical theories, methodologies, and applications of systems science and complexity science. It encourages fundamental research into complex systems and complexity and fosters cross-disciplinary approaches to elucidate the common mathematical methods that arise in natural, artificial, and social systems. Topics covered are: complex systems, systems control, operations research for complex systems, economic and financial systems analysis, statistics and data science, computer mathematics, systems security, coding theory and crypto-systems, other topics related to systems science.
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