{"title":"A New Method for Solving Nonlinear Partial Differential Equations Based on Liquid Time-Constant Networks","authors":"Jiuyun Sun, Huanhe Dong, Yong Fang","doi":"10.1007/s11424-024-3349-z","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>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.</p>","PeriodicalId":50026,"journal":{"name":"Journal of Systems Science & Complexity","volume":"1 1","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2024-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Systems Science & Complexity","FirstCategoryId":"1089","ListUrlMain":"https://doi.org/10.1007/s11424-024-3349-z","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
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.
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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