天元系列活动 | Neural Networks with Local Converging Inputs (NNLCI) for Solving PDEs: A General Approach with Orders-of-Magnitude Complexity Reduction and Minimal Training Data Requirements-国家天元数学东北中心

天元系列活动 | Neural Networks with Local Converging Inputs (NNLCI) for Solving PDEs: A General Approach with Orders-of-Magnitude Complexity Reduction and Minimal Training Data Requirements

报告人： Yingjie Liu,教授

所在单位： 佐治亚理工学院

报告地点： 中心校区数学楼第一报告厅

报告时间： 2025-06-03 14:00:00

报告简介：

This talk is based on a series of joint works with Zhen Chao, Harris Cobb, Haoxiang Huang,Weiming Ding, Hwi Lee, Tzu Jung Lee, Dexuan Xie, and Vigor Yang.

We have developed a neural network approach capable of accurately predicting both shock interactions and smooth regions of solutions to the Euler equations in 1D and 2D. For example, to predict the solution of the 1D Euler equations at a specific space-time location, the output of a neural network can be designed to provide the solution value at that location. However, if the input consists solely of a low-cost numerical solution patch within a local domain of dependence, the neural network lacks the ability to distinguish between inputs spanning a shock and those within a smooth region. Our approach leverages two numerical solutions from a converging sequence—computed using low-cost numerical schemes within the local domain of dependence of a given space-time location. These serve as the input for the neural network to generate high-fidelity solution at the location.

Despite the smeared nature of input solutions, the resulting output provides sharp approximations for solutions containing shocks and contact discontinuities.This method dramatically reduces computational complexity compared to fine-grid numerical simulations,achieving at least a two-order-of-magnitude reduction in 2D, with potential for even greater savings in higher dimensions due to its localized methodology. Moreover, the training data requirement remains minimal, as a single fine-grid simulation can generate hundreds or thousands of local samples for training. The method sustains strong generalization even when confronted with complex and pronounced singularities in the solutions. Beyond its efficiency,this approach naturally extends to complex domain geometries, enabling training on one domain geometry while facilitating predictions on another. We will also discuss extensions of this methodology to unstructured grids, electromagnetic wave scattering off curved conductors with corners, and the Poisson–Nernst–Planck ion channel model.