Randomized Neural Networks for Integro-Differential Equations with Application to Neutron Transport

2026-04-15

AI Summary

This work addresses the computational challenges in solving linear integro-differential equations involving nonlocal integral operators, where conventional methods suffer from high computational and memory costs due to dense coupling, and physics-informed neural networks encounter nonconvex optimization difficulties. The authors propose a mesh-free collocation method based on randomized artificial neural networks (RaNNs), which fixes the hidden-layer weights randomly and trains only the linear output-layer parameters, thereby reformulating the solution process as a convex least-squares problem. Leveraging the intrinsic global connectivity of RaNNs, the approach naturally accommodates nonlocal operators without resorting to sparse approximations, achieving efficient and stable solutions with extremely few trainable degrees of freedom. Numerical experiments on the steady-state neutron transport equation demonstrate that the method significantly reduces training costs compared to existing neural-network and deterministic baseline approaches while maintaining comparable accuracy, confirming its efficiency and robustness.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

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.

Problem

• integro-differential equations
• nonlocal operators
• computational cost
• neural networks
• neutron transport

Innovation

• Randomized Neural Networks
• Integro-Differential Equations
• Convex Training
• Mesh-Free Collocation
• Neutron Transport