This paper presents a numerical approach to the stochastic obstacle problem using the stochastic Galerkin (SG) method. Due to the low regularity of the solution, linear finite elements are employed in both the physical and random variable spaces. Properties of random fields and variational inequalities of the first kind are employed to establish the well-posedness of the problem. Finite element spaces are introduced to construct suitable approximation subspaces, and a comprehensive SG formulation is proposed to solve the stochastic obstacle problem. Well-posedness of the discrete formulation is shown and an optimal error estimate for the numerical solution in the H1-norm is derived. Numerical experiments validate the effectiveness of the SG method, showing that the expectation error converges at a rate of O(h) in the H1-norm, consistent with theoretical predictions.

Deep operator networks (DeepONets) represent a powerful class of data-driven methods for operator learning, demonstrating strong approximation capabilities for a wide range of linear and nonlinear operators. They have shown promising performance in learning operators that govern partial differential equations (PDEs), including diffusion-reaction systems and Burgers' equations. However, the accuracy of DeepONets is often constrained by computational limitations and optimization challenges inherent in training deep neural networks. Furthermore, the computational cost associated with training these networks is typically very high. To address these challenges, we leverage randomized neural networks (RaNNs), in which the parameters of the hidden layers remain fixed following random initialization. RaNNs compute the output layer parameters using the least-squares method, significantly reducing training time and mitigating optimization errors. In this work, we integrate DeepONets with RaNNs to propose RaNN-DeepONets, a hybrid architecture designed to balance accuracy and efficiency. Furthermore, to mitigate the need for extensive data preparation, we introduce the concept of physics-informed RaNN-DeepONets. Instead of relying on data generated through other time-consuming numerical methods, we incorporate PDE information directly into the training process. We evaluate the proposed model on three benchmark PDE problems: diffusion-reaction dynamics, Burgers' equation, and the Darcy flow problem. Through these tests, we assess its ability to learn nonlinear operators with varying input types. When compared to the standard DeepONet framework, RaNN-DeepONets achieves comparable accuracy while reducing computational costs by orders of magnitude. These results highlight the potential of RaNN-DeepONets as an efficient alternative for operator learning in PDE-based systems.

The efficient lean management of energy consumption offers technical support for energy-saving manufacturing of CNC equipment. However, intricate processing scenarios result in differences in the distribution of energy characteristics, which seriously reduces prediction accuracy. Additionally, existing machine learning (ML) techniques depend on numerous labeled samples exhibiting identical probability distributions, resulting a dramatic deterioration in the performance of conventional prediction methods. Therefore, this paper builds a metamodel based on physics-informed feature transfer learning, which is used to predict energy consumption in general milling scenarios. Firstly, combining different scenarios of CNC machining, Gaussian process (GP) model is employed to construct the variable mapping relationship of energy consumption in source domain and target domain. Next, feature transfer and energy prediction are carried out based on the shared feature processor of XGBoost. Meanwhile, Wasserstein distance (WD) and Shapley additive explanations (SHAP) are applied to measure and reveal the distribution distance and feature correlation between source domain and target domain. The overall prediction accuracy of the proposed model has been verified to be over 98.5% through experiments in target domain, demonstrating that the metamodel can be applied to general CNC milling process, thereby further promoting the development of green manufacturing.

Surface partial differential equations arise in numerous scientific and engineering applications. Their numerical solution on static and evolving surfaces remains challenging due to geometric complexity and, for evolving geometries, the need for repeated mesh updates and geometry or solution transfer. While neural-network-based methods offer mesh-free discretizations, approaches based on nonconvex training can be costly and may fail to deliver high accuracy in practice. In this work, we develop a randomized neural network (RaNN) method for solving PDEs on both static and evolving surfaces: the hidden-layer parameters are randomly generated and kept fixed, and the output-layer coefficients are determined efficiently by solving a least-squares problem. For static surfaces, we present formulations for parametrized surfaces, implicit level-set surfaces, and point-cloud geometries, and provide a corresponding theoretical analysis for the parametrization-based formulation with interface compatibility. For evolving surfaces with topology preserved over time, we introduce a RaNN-based strategy that learns the surface evolution through a flow-map representation and then solves the surface PDE on a space–time collocation set, avoiding remeshing. Extensive numerical experiments demonstrate broad applicability and favorable accuracy–efficiency performance on representative benchmarks.

Randomized neural networks (RaNNs) are attractive for partial differential equations (PDEs) because they replace expensive end-to-end training with a linear least-squares solve over randomized hidden features. Their practical performance, however, depends strongly on the sampling distribution of the hidden-layer parameters, which is usually chosen heuristically and problem by problem. This distribution sensitivity is a central bottleneck in randomized neural PDE solvers. In this work, we propose Adaptive-Distribution Randomized Neural Networks (AD-RaNN), a framework that promotes randomized feature generation from a fixed heuristic choice to a low-dimensional adaptive optimization problem. Instead of training all hidden weights and biases, AD-RaNN parameterizes the hidden-feature sampling distribution by a low-dimensional vector p and optimizes only p, thereby preserving the least-squares structure of RaNNs while reducing manual distribution tuning. The method uses a two-stage strategy: ridge-regularized reduced training for stable distribution-parameter optimization, followed by an unregularized least-squares refit for final solution recovery. We develop two adaptive mechanisms, PDE-Driven Adaptive Distribution (PDAD) and Data-Driven Adaptive Distribution (DDAD), and deploy them in space-time solvers, discrete-time solvers, and operator-learning models. We also incorporate an adaptive layer-growth enhancement for localized structures. For the reduced optimization problem, we establish well-posedness of the reduced objectives, consistency of ridge-regularized minimizers, an efficient gradient formula, and a practical lower-bound estimate for the ridge parameter. Numerical experiments on benchmark problems show that AD-RaNN provides an effective distribution-level adaptation mechanism, reduces reliance on hand-crafted hidden-feature distributions, and achieves strong empirical accuracy.

The low efficiency and accuracy are the bottleneck problems that constrain neural network-based approaches from solving multi-scale problems. In this study, an innovative higher-order multi-scale method enhanced by physics-informed randomized neural network (HOMS-PIRNN) is proposed to efficiently and accurately compute transient thermal problems of composite materials with discontinuous and high-contrast material parameters, which inherits the respective high efficiency, accuracy, mesh-free and space–time advantages of higher-order multi-scale approach and PIRNN. To address the large complexity and Frequency Principle of multi-scale problems, higher-order multi-scale approach is employed to decouple and reduce complicated multi-scale thermal problem of heterogeneous materials as simpler macroscopic homogenized and small-scale lower-order and higher-order microscopic cell problems. Then, by using macroscopic homogenized and microscopic cell problems as new physical constraints, PIRNN are designed to efficiently solve homogenized equations and microscopic unit cell functions, for further assembling higher-order multi-scale asymptotic solutions for multi-scale heat problems via automatic differentiation. Moreover, the error estimate of the proposed HOMS-PIRNN method is rigorously demonstrated under some hypotheses. The efficiency and accuracy of the proposed HOMS-PIRNN are validated through multi-scale thermal simulations of high-contrast composites, three-dimensional composites, and porous materials. The efficient HOMS-PIRNN computational framework with strong potential for large-scale heat simulations of complex composites.

We study operator learning for random obstacle-to-solution maps arising from elliptic variational inequalities with finite-band self-affine random obstacle fields. Instead of introducing an explicit truncated stochastic parametrization of the random input, we learn the map directly from sampled obstacle realizations on a fixed grid. This problem is challenging because the solution is governed not only by the obstacle field itself, but also by the induced contact set and free-boundary geometry. We introduce a post-training least-squares readout refit for the Fourier neural operator (FNO). After the FNO is trained end to end, its nonlinear backbone is frozen and the final affine readout is recomputed by solving the induced linear least-squares problem over all training samples and grid points. The refit yields the empirical squared-error optimal readout for the learned frozen features while leaving the nonlinear representation unchanged. We compare vanilla DeepONet, POD-DeepONet, a two-stage DeepONet baseline, FNO, and FNO with least-squares readout refit (FNO-LS) on two obstacle ensembles with different amplitude levels. Numerical results show that FNO-LS achieves the strongest overall performance among the tested models, particularly for higher-amplitude obstacles with more complex contact geometry. The method improves average field accuracy, contact-set recovery, and obstacle-violation metrics at low additional cost, especially when the FNO backbone is informative but not fully converged. These results suggest that least-squares readout refit is a simple and effective post-training enhancement for learning random obstacle-to-solution maps.

The incompressible magnetohydrodynamic (MHD) equations are fundamental in many scientific and engineering applications. However, their strong nonlinearity and dual divergence-free constraints make them highly challenging for conventional numerical solvers. To overcome these difficulties, we propose a Structure-Preserving Randomized Neural Network (SP-RaNN) that automatically and exactly satisfies the divergence-free conditions. Unlike deep neural network (DNN) approaches that rely on expensive nonlinear and nonconvex optimization, SP-RaNN reformulates the training process into a linear least-squares system, thereby eliminating nonconvex optimization. The method linearizes the governing equations through Picard or Newton iterations, discretizes them at collocation points within the domain and on the boundaries using finite-difference schemes, and solves the resulting linear system via a linear least-squares procedure. By design, SP-RaNN preserves the intrinsic mathematical structure of the equations within a unified space-time framework, ensuring both stability and accuracy. Numerical experiments on the Navier-Stokes, Maxwell, and MHD equations demonstrate that SP-RaNN achieves higher accuracy, faster convergence, and exact enforcement of divergence-free constraints compared with both traditional numerical methods and DNN-based approaches. This structure-preserving framework provides an efficient and reliable tool for solving complex PDE systems while rigorously maintaining their underlying physical laws.

Repeatedly solving parametric PDEs is essential for uncertainty quantification, design optimization and inverse problems, but conventional neural operators require expensive non-convex training. We introduce PCA–RaNN, a randomized latent neural operator that combines PCA-based dimensionality reduction with fixed random features and a closed-form least-squares readout. It recasts latent operator learning as fixed-feature linear regression, reducing training time by one to three orders of magnitude across benchmarks while maintaining competitive accuracy. We introduce an energy-matched scaling rule and a lightweight two-parameter BFGS refinement to correct suboptimal feature scales. Ensemble averaging reduces predictive variance. On Burgers, Darcy, Navier–Stokes and backward heat equation benchmarks, PCA–RaNN provides a favorable speed–accuracy trade-off against operator-learning baselines. The ensemble supports split-conformal prediction intervals, and the linear readout enables rapid online adaptation via recursive least squares without retraining hidden features. This provides an efficient, uncertainty-aware surrogate for many-query scientific workflows.

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.

Partial differential equations on unbounded domains are challenging because the exterior region must be represented without excessive truncation error. Truncation-based methods often require problem-dependent artificial boundary conditions, while global spectral bases may be inefficient for localized structures, irregular geometries, or solutions with different near-field and far-field behaviors. We propose a domain-decomposed randomized neural network framework for such problems. Different randomized subnetworks are assigned to different spatial regimes: a near-field subnetwork captures local and geometric features, whereas a far-field subnetwork represents exterior decay. The subnetworks are coupled by boundary and interface conditions, and only the output-layer coefficients are solved from linear least-squares systems arising from Petrov–Galerkin or collocation formulations. We develop a Petrov–Galerkin method for semi-unbounded elliptic problems and a collocation method for fully unbounded, perforated, and time-dependent problems. A conditional bounded-parameter approximation result is proved in a broken Sobolev norm, together with an error decomposition covering approximation, empirical-consistency/quadrature, and least-squares optimization errors. Numerical experiments for Poisson and time-dependent Schrödinger equations demonstrate the accuracy and flexibility of the proposed method.

The state and adjoint state variables are approximated within a unified neural network framework, avoiding conventional mesh-based discretization. The control variable is then obtained from the adjoint state via the optimality condition. Numerical results show that the RaNN-based approach can achieve satisfactory accuracy with a relatively small number of degrees of freedom and is capable of handling pointwise control constraints. Moreover, for problems defined on non-smooth domains, such as the circular sector domain with a reentrant corner, suitable singular treatments are incorporated to enhance the approximation quality near corner singularities. Several two- and three-dimensional numerical examples are presented to demonstrate the effectiveness and robustness of the RaNN method.

We develop a structure-oriented randomized neural network framework, termed SO-RaNN, for the Poisson-Nernst-Planck (PNP) system and the Poisson-Nernst-Planck-Navier-Stokes (PNP-NS) system. The decoupled linearized subproblems are solved iteratively by randomized neural networks in a space-time framework. For the concentration variables, a pointwise cut-off is used to enforce positivity at the value level, and discrete mass-scaling factors are computed at selected correction instants and interpolated in time, so as to ensure exact mass matching at those instants and to promote approximate mass preservation between them. To introduce an auxiliary discrete dissipation mechanism, we further employ an SAV-type post-processing correction, which yields monotonicity of the SAV auxiliary variable under the ideal SAV update. For the PNP-NS system, a structure-preserving randomized neural network (SP-RaNN) is used for the velocity field, so that the velocity approximation satisfies the incompressibility constraint pointwise by construction. On the theoretical side, we derive residual-based estimates for the raw, uncorrected RaNN solvers of the linearized subproblems, formulate a conditional local-in-time convergence result for the raw outer Picard iteration of the PNP system, and analyze the value-level positivity correction together with the mass-correction and SAV post-processing steps. For the PNP-NS system, we establish an approximation result for the SP-RaNN space and provide a conditional error statement for the corresponding linearized Oseen-type problem. Numerical experiments demonstrate approximation accuracy in the source-driven manufactured tests and illustrate the intended value-level positivity correction, selected-time mass matching, computed free-energy curves based on the final gauge-fixed potential, and divergence-free approximation in benchmark tests.

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.

The transition of CNC machining toward digitalization and low-carbon manufacturing is essential for the advancement of intelligent production. However, conventional parameter configuration methods fail to balance efficiency and sustainability. To overcome this limitation, this study proposes an intelligent optimization framework that integrates digital twin (DT) technology with multi-objective optimization. A multi-level virtual machine tool model is established to enable operational condition mapping and structural response modeling of key machining parameters. A Simulation Augmentation Collaboration Mechanism (SACM) is further introduced, in which the DT generates high-fidelity distribution information to guide a Wasserstein Generative Adversarial Network with Gradient Penalty (WGAN-GP) in producing realistic samples under critical operating conditions. These augmented data iteratively refine the model, significantly enhancing predictive generalization. An Improved Meta-Learning-Optimized XGBoost (IMeta-XGBoost) model is then established to predict three performance indicators: spindle energy consumption, specific cutting energy, and machining noise. A Predicted-Fitness-Guided Multi-Objective Deep Q-Network (PF-MO-DQN) is then employed for global optimization, followed by entropy-weighted TOPSIS to determine the optimal machining parameters experimental validation demonstrates reductions of 8.95% in spindle energy consumption, 18.03% in specific cutting energy, and 10.15% in machining noise, confirming significant improvements in energy efficiency, productivity, and noise mitigation. This work provides a robust and scalable approach for multi-objective optimization in complex machining environments.

In this paper, we develop a Discontinuous Galerkin (DG) method for solving H(curl)-elliptic hemivariational inequalities. By selecting an appropriate numerical flux, we construct an Interior Penalty Discontinuous Galerkin (IPDG) scheme. A comprehensive numerical analysis of the IPDG method is conducted, addressing key aspects such as consistency, boundedness, stability, and the existence, uniqueness, uniform boundedness of the numerical solutions. Building on these properties, we establish a priori error estimates, demonstrating the optimal convergence order of the numerical solutions under suitable solution regularity assumptions. Finally, a numerical example is presented to illustrate the theoretically predicted convergence order and to show the effectiveness of the proposed method.

We study the numerical solution of an elastic-rigid obstacle problem. The problem describes the equilibrium position of an elastic membrane constrained by a composite obstacle comprising a rigid substrate and a deformable soft layer that permits limited penetration. During contact, the obstacle exerts a normal counter-pressure, resulting in two types of contact regions: contact with the soft layer and contact with the rigid substrate. The nonlinear coupling between the normal reaction pressure and the degree of mutual interpenetration, along with the two contact regions with free boundaries, poses significant difficulties for traditional numerical discretization methods. In this work, the Virtual Element Method (VEM) is employed to discretize the elastic-rigid obstacle problem. To handle the nonlinear coupling, different strategies are adopted for linear and quadratic elements. For linear elements, the standard virtual element space is employed in order to preserve the Maximum Principle, together with an approximate L-2-projection as the exact one is not directly computable from the degrees of freedom. For quadratic elements, an enhanced virtual element space is introduced, which allows the exact computation of the L-2-projection. We prove that the linear elements achieve the optimal convergence rate, while a convergence rate Theta(h(3/2-epsilon)) is obtained for quadratic elements under suitable regularity assumptions. Numerical experiments on general polygonal meshes confirm the theoretical results.

AI has become an essential tool in many areas of scientific discovery. However, its robustness, particularly in high-stakes scientific applications, remains insufficiently understood. While adversarial vulnerabilities have been widely studied in image recognition, their implications in scientific disciplines, where physical constraints and numerical stability are critical, have not been thoroughly explored. This work presents a systematic evaluation of AI model fragility across five scientific domains: climate modeling, quantum chemistry, fluid dynamics, quantum chromodynamics, and wireless communications. We demonstrate that small, structured perturbations can destabilize predictions in all of these domains, even in models that achieve high accuracy under normal conditions. Furthermore, the failure patterns vary across domains, suggesting that these vulnerabilities are domain specific. Our findings underscore the importance of assessing AI robustness in scientific contexts and offer preliminary insights into how to design more resilient and reliable AI systems for scientific applications.

Accurate modeling of complex physical problems, such as fluid-structure interaction, requires multiphysics coupling across the interface, which often has intricate geometry and dynamic boundaries. Conventional numerical methods face challenges in handling interface conditions. Deep neural networks offer a mesh-free and flexible alternative, but they suffer from drawbacks such as time-consuming optimization and local optima. In this paper, we propose a mesh-free approach based on Randomized Neural Networks (RaNNs) and finite difference methods (FDM), which avoid optimization solvers during training, making them more efficient than traditional deep neural networks. Our approach, called Local Randomized Neural Networks with finite difference methods (LRaNN-FDM), uses different RaNNs to approximate solutions in different subdomains. We discretize the interface problem into a linear system at randomly sampled points across the domain, boundary, and interface using a finite difference scheme, and then solve it by a least-square method. Unlike automatic differentiation for partial derivative calculations, the finite difference approach offers significantly faster computation. For time-dependent interface problems, we use a space-time approach based on LRaNNs. We show the effectiveness and robustness of the LRaNN-FDM through numerical examples of elliptic and parabolic interface problems. We also demonstrate that our approach can handle high-dimension interface problems. Compared to conventional numerical methods, our approach achieves higher accuracy with fewer degrees of freedom, eliminates the need for complex interface meshing and fitting, and significantly reduces training time, outperforming deep neural networks.

This work is concerned with the mathematical and numerical analyses of the Navier-Stokes equations driven by threshold power law slip boundary condition. The tangential shear belongs to the sub-differential of a power law function, giving rise to quasi-variational inequalities. We construct a unique weak solution and formulate a mixed finite element approximation to the system using conforming finite element spaces. The unique solvability of the finite element approximation is investigated and an abstract convergence result is established. We derive a-priori error estimate by providing a bound on the error between the weak solution and its finite element approximation in terms of the best approximation error from the finite element spaces. We introduce a Lagrange multiplier and propose a Uzawa-type iterative scheme to solve the nonlinear problem resulting from the finite element approximation. Finally, numerical results are provided to validate the theoretical findings.