解算器
蒙特卡罗方法
计算机科学
不确定度量化
概率逻辑
离散化
算法
代表(政治)
数学优化
人工智能
机器学习
数学
数学分析
统计
政治
政治学
法学
程序设计语言
作者
Rui Zhang,Qiang Meng,Rongchan Zhu,Yue Wang,Wenlei Shi,Shihua Zhang,Zhong‐Qi Ma,Tie-Yan Liu
出处
期刊:Cornell University - arXiv
日期:2023-01-01
被引量:1
标识
DOI:10.48550/arxiv.2302.05104
摘要
In scenarios with limited available or high-quality data, training the function-to-function neural PDE solver in an unsupervised manner is essential. However, the efficiency and accuracy of existing methods are constrained by the properties of numerical algorithms, such as finite difference and pseudo-spectral methods, integrated during the training stage. These methods necessitate careful spatiotemporal discretization to achieve reasonable accuracy, leading to significant computational challenges and inaccurate simulations, particularly in cases with substantial spatiotemporal variations. To address these limitations, we propose the Monte Carlo Neural PDE Solver (MCNP Solver) for training unsupervised neural solvers via the PDEs' probabilistic representation, which regards macroscopic phenomena as ensembles of random particles. Compared to other unsupervised methods, MCNP Solver naturally inherits the advantages of the Monte Carlo method, which is robust against spatiotemporal variations and can tolerate coarse step size. In simulating the random walk of particles, we employ Heun's method for the convection process and calculate the expectation via the probability density function of neighbouring grid points during the diffusion process. These techniques enhance accuracy and circumvent the computational memory and time issues associated with Monte Carlo sampling, offering an improvement over traditional Monte Carlo methods. Our numerical experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstrate significant improvements in accuracy and efficiency compared to other unsupervised baselines. The source code will be publicly available at: https://github.com/optray/MCNP.
科研通智能强力驱动
Strongly Powered by AbleSci AI