Haoning Dang

Haoning Dang, Fei Wang, Yifan Chen et al.

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.

Haoning Dang, Shi Jin, Fei Wang

This paper proposes an Adaptive-Growth Randomized Neural Network (AG-RaNN) method for computing multivalued solutions of nonlinear first-order PDEs with hyperbolic characteristics, including quasilinear hyperbolic balance laws and Hamilton--Jacobi equations. Such solutions arise in geometric optics, seismic waves, semiclassical limit of quantum dynamics and high frequency limit of linear waves, and differ markedly from the viscosity or entropic solutions. The main computational challenges lie in that the solutions are no longer functions, and become union of multiple branches, after the formation of singularities. Level-set formulations offer a systematic alternative by embedding the nonlinear dynamics into linear transport equations posed in an augmented phase space, at the price of substantially increased dimensionality. To alleviate this computational burden, we combine AG-RaNN with an adaptive collocation strategy that concentrates samples in a tubular neighborhood of the zero level set, together with a layer-growth mechanism that progressively enriches the randomized feature space. Under standard regularity assumptions on the transport field and the characteristic flow, we establish a convergence result for the AG-RaNN approximation of the level-set equations. Numerical experiments demonstrate that the proposed method can efficiently recover multivalued structures and resolve nonsmooth features in high-dimensional settings.

Jun-Jie Zhang, Jiahao Song, Xiu-Cheng Wang et al.

We uncover a phenomenon largely overlooked by the scientific community utilizing AI: neural networks exhibit high susceptibility to minute perturbations, resulting in significant deviations in their outputs. Through an analysis of five diverse application areas -- weather forecasting, chemical energy and force calculations, fluid dynamics, quantum chromodynamics, and wireless communication -- we demonstrate that this vulnerability is a broad and general characteristic of AI systems. This revelation exposes a hidden risk in relying on neural networks for essential scientific computations, calling further studies on their reliability and security.

Huibo Xu, Likang Wu, Xianquan Wang et al.

Time series forecasting is a fundamental task with broad applications, yet conventional methods often treat data as discrete sequences, overlooking their origin as noisy samples of continuous processes. Crucially, discrete noisy observations cannot uniquely determine a continuous function; instead, they correspond to a family of plausible functions. Mathematically, time series can be viewed as noisy observations of a continuous function family governed by a shared probability measure. Thus, the forecasting task can be framed as learning the transition from the historical function family to the future function family. This reframing introduces two key challenges: (1) How can we leverage discrete historical and future observations to learn the relationships between their underlying continuous functions? (2) How can we model the transition path in function space from the historical function family to the future function family? To address these challenges, we propose NeuTSFlow, a novel framework that leverages Neural Operators to facilitate flow matching for learning path of measure between historical and future function families. By parameterizing the velocity field of the flow in infinite-dimensional function spaces, NeuTSFlow moves beyond traditional methods that focus on dependencies at discrete points, directly modeling function-level features instead. Experiments on diverse forecasting tasks demonstrate NeuTSFlow's superior accuracy and robustness, validating the effectiveness of the function-family perspective.