Chrome Extension
WeChat Mini Program
Use on ChatGLM
VIPCreated with Sketch.

    • Deep FBSDE Neural Networks for Solving Incompressible Navier-Stokes Equation and Cahn-Hilliard Equation

    Yangtao Deng,Qiaolin He

    arXiv (Cornell University)（2024）

    Sichuan University School of Mathematics | Sichuan University Qiaolin He School of Mathematics

    Cited 0|Views13
    Abstract
    Efficient algorithms for solving high-dimensional partial differentialequations (PDEs) has been an exceedingly difficult task for a long time, due tothe curse of dimensionality. We extend the forward-backward stochastic neuralnetworks (FBSNNs) which depends on forward-backward stochastic differentialequation (FBSDE) to solve incompressible Navier-Stokes equation. ForCahn-Hilliard equation, we derive a modified Cahn-Hilliard equation from awidely used stabilized scheme for original Cahn-Hilliard equation. Thisequation can be written as a continuous parabolic system, where FBSDE can beapplied and the unknown solution is approximated by neural network. Also ourmethod is successfully developed to Cahn-Hilliard-Navier-Stokes (CHNS)equation. The accuracy and stability of our methods are shown in many numericalexperiments, specially in high dimension.
    More
    Translated text
    Key words
    Partial Differential Equations
    PDF
    View via Publisher
    Bibtex
    AI Read Science
    AI Summary
    AI Summary is the key point extracted automatically understanding the full text of the paper, including the background, methods, results, conclusions, icons and other key content, so that you can get the outline of the paper at a glance.
    Example
    Background
    Key content
    Introduction
    Methods
    Results
    Related work
    Fund
    Key content
    • Pretraining has recently greatly promoted the development of natural language processing (NLP)
    • We show that M6 outperforms the baselines in multimodal downstream tasks, and the large M6 with 10 parameters can reach a better performance
    • We propose a method called M6 that is able to process information of multiple modalities and perform both single-modal and cross-modal understanding and generation
    • The model is scaled to large model with 10 billion parameters with sophisticated deployment, and the 10 -parameter M6-large is the largest pretrained model in Chinese
    • Experimental results show that our proposed M6 outperforms the baseline in a number of downstream tasks concerning both single modality and multiple modalities We will continue the pretraining of extremely large models by increasing data to explore the limit of its performance
    Try using models to generate summary,it takes about 60s
    Must-Reading Tree
    Example
    Generate MRT to find the research sequence of this paper
    Data Disclaimer
    The page data are from open Internet sources, cooperative publishers and automatic analysis results through AI technology. We do not make any commitments and guarantees for the validity, accuracy, correctness, reliability, completeness and timeliness of the page data. If you have any questions, please contact us by email: report@aminer.cn
    Chat Paper
    【要点】：本文提出了基于向前-向后随机微分方程（FBSDE）的深度神经网络（FBSNNs）方法，用于高效解决高维不可压缩Navier-Stokes方程和Cahn-Hilliard方程，以及Cahn-Hilliard-Navier-Stokes（CHNS）方程，实现了算法在高维情况下的稳定性和准确性。
    

    【方法】：作者通过扩展FBSNNs，将FBSDE方法应用于解决不可压缩Navier-Stokes方程，并对Cahn-Hilliard方程进行了改进，导出了一个适用于神经网络的连续抛物型系统。
    

    【实验】：在多个数值实验中，特别是在高维情况下，验证了所提方法在解决Navier-Stokes方程和Cahn-Hilliard方程时的准确性和稳定性，但论文中未明确提及所使用的数据集名称。
    去 AI 文献库 对话