随机神经网络用于积分-微分方程及其在中子输运中的应用
Randomized Neural Networks for Integro-Differential Equations with Application to Neutron Transport
- School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）
- School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University（西安交通大学数学与统计学院及电力工程多相流国家重点实验室）
- School of Energy and Power Engineering, Xi’an Jiaotong University（西安交通大学能源与动力工程学院）
- Nuclear Power Institute of China, State Key Laboratory of Advanced Nuclear Energy Technology（中国核工业集团核电院，先进核能技术国家重点实验室）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出随机神经网络RaNNs，用于求解线性积分-微分方程，通过全局支持的随机特征实现内在密集性，减少计算成本并提高稳定性，应用于中子输运方程展示其高效性。
AI中文摘要：
积分-微分方程出现在运输、动力学理论、辐射传输和多物理场建模中，其中非局部积分算子在相空间中耦合解。此类非局部性常导致确定性离散化中密集耦合块，增加计算成本和内存使用，而物理信息神经网络可能面临昂贵的非凸训练和超参数敏感性。本文提出随机神经网络（RaNNs）作为线性积分-微分方程的无网格插值框架。由于RaNN近似通过全局支持的随机特征内在密集，非局部积分算子不引入额外的稀疏性损失，而近似解仍可用相对较少的可训练自由度表示。通过随机固定隐藏层参数并仅求解线性输出权重，训练过程减少为输出系数的凸最小二乘问题，实现稳定高效的优化。作为代表性应用，将所提框架应用于稳态中子输运方程，一个高维线性积分-微分模型，包含散射积分和多样的边界条件。大量数值实验表明，在报告的测试设置中，RaNN方法在准确性上具有竞争力，但训练成本显著低于所选神经网络和确定性基线，突显RaNNs作为非局部线性算子数值模拟的稳健且高效的替代方案。
英文摘要：
Integro-differential equations arise in a wide range of applications, including transport, kinetic theory, radiative transfer, and multiphysics modeling, where nonlocal integral operators couple the solution across phase space. Such nonlocality often introduces dense coupling blocks in deterministic discretizations, leading to increased computational cost and memory usage, while physics-informed neural networks may suffer from expensive nonconvex training and sensitivity to hyperparameter choices. In this work, we present randomized neural networks (RaNNs) as a mesh-free collocation framework for linear integro-differential equations. Because the RaNN approximation is intrinsically dense through globally supported random features, the nonlocal integral operator does not introduce an additional loss of sparsity, while the approximate solution can still be represented with relatively few trainable degrees of freedom. By randomly fixing the hidden-layer parameters and solving only for the linear output weights, the training procedure reduces to a convex least-squares problem in the output coefficients, enabling stable and efficient optimization. As a representative application, we apply the proposed framework to the steady neutron transport equation, a high-dimensional linear integro-differential model featuring scattering integrals and diverse boundary conditions. Extensive numerical experiments demonstrate that, in the reported test settings, the RaNN approach achieves competitive accuracy while incurring substantially lower training cost than the selected neural and deterministic baselines, highlighting RaNNs as a robust and efficient alternative for the numerical simulation of nonlocal linear operators.
