哈尔滨工业大学（深圳）学术讲座

演讲人Speaker：    王飞

题目Title:Random Neural Network Methods for Partial Differential Equations

时间Date：2026年9月23日    Time：15：00 -15：40

地点Venue：线上腾讯会议，会议号839-763-365

内容摘要Abstract：

Numerical solution of partial differential equations is a central problem in scientific computing. Traditional numerical methods rest on a rigorous mathematical foundation and hold clear advantages in accuracy, stability, and structure preservation; yet in high-dimensional, geometrically complex, multiscale-coupled, interface, and many-query parametric settings, they still face bottlenecks such as strong mesh dependence, a large number of degrees of freedom, high cost of repeated solves, and limited capacity for rapid adaptation. Neural network methods offer flexible representation capability, but they typically rely on nonlinear, nonconvex optimization and therefore suffer from high training cost, insufficient stability, and weak error controllability.

This report focuses on random neural network (RaNN) methods and presents several advances in their application to PDE solving and operator learning. A random neural network fixes the random parameters of the hidden layers and solves only for the output-layer weights, thereby reducing training to a linear system or a least-squares problem, and so combines the expressive power of neural networks with the analyzability of traditional numerical methods. Building on this idea, we have developed RaNN–Petrov–Galerkin, local RaNN–DG, local RaNN–HDPG, local RaNN–finite difference, adaptively grown RaNN, adaptive distribution RaNN, Schwarz-preconditioned RaNN, multiscale RaNN, near–far-field coupled RaNN for unbounded domains, and random neural operator methods, achieving a deep integration of random feature spaces with weak formulations, domain decomposition, DG schemes, hybridization structures, finite differences, interface conditions, and operator learning frameworks.

Numerical results show that RaNN methods attain high-accuracy solutions with relatively few degrees of freedom, and that they exhibit promising efficiency, stability, and scalability in complex-interface, high-dimensional, multiscale, structure-preserving, unbounded-domain, and parametric operator learning problems. Overall, RaNN methods offer a promising new route toward fusing traditional numerical analysis with modern machine learning and toward constructing learned scientific computing methods that are error-controllable, structure-preserving, efficient, and scalable.

个人简介(About the speaker):

王飞，西安交通大学数学与统计学院教授、博士生导师，Communications in Nonlinear Science and Numerical Simulation副主编，陕西省青年百人入选者、西安交通大学青年拔尖人才。研究领域涵盖科学计算与机器学习，主要研究兴趣包括偏微分方程数值解、深度学习与算子学习等。主持国家自然科学基金重大研究计划培育项目1项、面上项目2项、青年基金1项，参与国家重点研发计划项目1项；已在国际SCI期刊发表论文七十篇。近年来，王教授聚焦神经网络方法与传统数值方法的融合，提出了随机神经网络Petrov–Galerkin方法、局部随机神经网络DG方法、局部随机神经网络HDPG方法以及自适应生长随机神经网络方法等一系列新方法。同时，他还发展了基于随机神经网络的算子学习方法，在提升参数化PDE预测效率的同时，为算子网络训练提供了有效的加速机制。