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Randomized Neural Networks for Integro-Differential Equations with Application to Neutron Transport
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DOI：10.1016/j.cpc.2026.110300.png)
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Abstract
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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 propose a collocation randomized neural network framework for linear integro-differential equations, where the neural network trial space is mesh-independent in the sense that it does not rely on element connectivity or mesh-dependent basis functions. 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. Numerical experiments in the reported test settings show that the RaNN approach achieves competitive accuracy and reduces the optimization cost compared with the selected PINN baselines, highlighting RaNNs as a robust and efficient alternative for the numerical simulation of nonlocal linear operators.
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