Neural operators (NO) are discretization invariant deep learning methods with functional output and can approximate any continuous operator. NO has demonstrated the superiority of solving partial differential equations (PDEs) over other deep learning methods. However, for the widely used Fourier neural operator (FNO), the spatial domain of its input function needs to be identical to its output, i.e., FNO fails to approximate the map from boundary conditions to PDE solutions, which limits its applicability. To address this issue, we propose a novel framework called resolution-invariant deep operator (RDO) that decouples the spatial domain of the input and output. RDO is motivated by the Deep operator network (DeepONet) and it does not require retraining the network when the input/output is changed compared with DeepONet. RDO takes functional input and its output is also functional so that it keeps the resolution invariant property of NO. It can also resolve PDEs with complex geometries whereas FNO fails. Various numerical experiments demonstrate the advantage of our method over DeepONet and FNO.

Conformal prediction (CP) is a robust statistical framework that generates prediction intervals or sets with guaranteed coverage probability, addressing the challenge of quantifying predictive uncertainty in deep learning. Despite advancements in deep learning architectures and datasets, reliable uncertainty estimation remains elusive, making CP increasingly vital. This paper introduces TorchCP, a PyTorch-native library designed to integrate state-of-the-art CP algorithms into deep learning tasks, including classification, regression, graph neural networks, and large language models. TorchCP offers a comprehensive suite of advanced methodologies, a modular design for easy customization, and full GPU-accelerated scalability. Released under the LGPL-3.0 license, TorchCP has gained widespread adoption with over 12,582 PyPi downloads. It is supported by approximately 16,132 lines of code, 564 unit tests achieving 100% coverage, and comprehensive documentation. By bridging statistics and computer science, TorchCP empowers researchers and practitioners to advance conformal prediction in diverse deep learning applications.

This paper focuses on discussing Newton's method and its hybrid with machine learning for the steady state Navier-Stokes Darcy model discretized by mixed element methods. First, a Newton iterative method is introduced for solving the relative discretized problem. It is proved technically that this method converges quadratically with the convergence rate independent of the finite element mesh size, under certain standard conditions. Later on, a deep learning algorithm is proposed for solving this nonlinear coupled problem. Following the ideas of an earlier work by Huang, Wang and Yang (2020), an Int-Deep algorithm is constructed by combining the previous two methods so as to further improve the computational efficiency and robustness. A series of numerical examples are reported to show the numerical performance of the proposed methods.

Conformal prediction is a statistical framework that generates prediction sets containing ground-truth labels with a desired coverage guarantee. The predicted probabilities produced by machine learning models are generally miscalibrated, leading to large prediction sets in conformal prediction. To address this issue, we propose a novel algorithm named $\textit{Sorted Adaptive Prediction Sets}$ (SAPS), which discards all the probability values except for the maximum softmax probability. The key idea behind SAPS is to minimize the dependence of the non-conformity score on the probability values while retaining the uncertainty information. In this manner, SAPS can produce compact prediction sets and communicate instance-wise uncertainty. Extensive experiments validate that SAPS not only lessens the prediction sets but also broadly enhances the conditional coverage rate of prediction sets.

Conformal prediction is an emerging technique for uncertainty quantification that constructs prediction sets guaranteed to contain the true label with a predefined probability. Previous works often employ temperature scaling to calibrate classifiers, assuming that confidence calibration benefits conformal prediction. However, the specific impact of confidence calibration on conformal prediction remains underexplored. In this work, we make two key discoveries about the impact of confidence calibration methods on adaptive conformal prediction. Firstly, we empirically show that current confidence calibration methods (e.g., temperature scaling) typically lead to larger prediction sets in adaptive conformal prediction. Secondly, by investigating the role of temperature value, we observe that high-confidence predictions can enhance the efficiency of adaptive conformal prediction. Theoretically, we prove that predictions with higher confidence result in smaller prediction sets on expectation. This finding implies that the rescaling parameters in these calibration methods, when optimized with cross-entropy loss, might counteract the goal of generating efficient prediction sets. To address this issue, we propose Conformal Temperature Scaling (ConfTS), a variant of temperature scaling with a novel loss function designed to enhance the efficiency of prediction sets. This approach can be extended to optimize the parameters of other post-hoc methods of confidence calibration. Extensive experiments demonstrate that our method improves existing adaptive conformal prediction methods in classification tasks, especially with LLMs.

A robust nonconforming mixed finite element method is developed for a strain gradient elasticity (SGE) model. In two and three dimensional cases, a lower order $C^0$-continuous $H^2$-nonconforming finite element is constructed for the displacement field through enriching the quadratic Lagrange element with bubble functions. This together with the linear Lagrange element is exploited to discretize a mixed formulation of the SGE model. The robust discrete inf-sup condition is established. The sharp and uniform error estimates with respect to both the small size parameter and the Lamé coefficient are achieved, which is also verified by numerical results. In addition, the uniform regularity of the SGE model is derived under two reasonable assumptions.

Some estimates on virtual element methods (VEMs), including inverse inequalities, norm equivalence, and interpolation error estimates, are developed for star-shaped polyhedral meshes whose faces admit virtual regular and quasi-uniform triangulations. This mesh regularity covers the usual ones used for theoretical analysis of VEMs, and the proofs are carried out in a straightforward way.

This paper is concerned with a hybrid method for three-dimensional semi -linear elliptic equations, constructed by combining the ideas presented in [Huang et al., J. Comput. Phys. 419 (2020)] and [Zhang et al., Comput. Math. Appl. 80 (2020)]. The convergence rate analysis indicates that the method converges rapidly. Numerical ex-amples support the theoretical results and show that the method proposed outperforms the purely deep learning-based and traditional iterative methods.

This paper is devoted to a fourth-order hemivariational inequality for a Kirchhoff plate problem. A solution existence and uniqueness result is proved for the hemivariational inequality through the analysis of a corresponding minimization problem. A nonconforming virtual element method is developed to solve the hemivariational inequality. An optimal order error estimate in a broken $$H^2$$ -norm is derived for the virtual element solutions under appropriate solution regularity assumptions. The discrete problem can be formulated as an optimization problem for a difference of two convex (DC) functions and a convergent algorithm is used to solve it. Computer simulation results on a numerical example are reported, providing numerical convergence orders that match the theoretical prediction.

A residual-type a posteriori error estimation is developed for a C-1-conforming virtual element method (VEM) to solve a Kirchhoff plate bending problem. To derive the reliability and efficiency of the a posteriori error bound, the inverse inequalities and norm equivalence are developed over the underlying C-1-conforming virtual element space, and a weak interpolation operator together with its error estimates is given as well. As an outcome of the error estimator, an adaptive VEM is introduced by means of the mesh refinement strategy with the one-hanging-node rule. Numerical results on various benchmark tests confirm the robustness of the proposed error estimator and show the efficiency of the resulting adaptive VEM.

This paper introduces a deep learning method for solving an elliptic hemivariational inequality (HVI). In this method, an expectation minimization problem is first formulated based on the variational principle of underlying HVI, which is solved by stochastic optimization algorithms using three different training strategies for updating network parameters. The method is applied to solve two practical problems in contact mechanics, one of which is a frictional bilateral contact problem and the other of which is a frictionless normal compliance contact problem. Numerical results show that the deep learning method is efficient in solving HVIs and the adaptive mesh-free multigrid algorithm can provide the most accurate solution among the three learning methods.

This paper is devoted to the numerical solution of a fourth-order elliptic variational inequality of the first kind by the virtual element method (VEM). The variational inequality models an obstacle problem for the Kirchhoff-Love plate. Both conforming and fully nonconforming VEMs are studied to solve the fourth-order elliptic variational inequality. Optimal order error estimates are derived in the discrete energy norm, under certain solution regularity assumptions. The primal-dual active algorithm is applied to solve the discrete problems. Numerical examples are reported to show the performance of the numerical methods and to illustrate the convergence orders of the numerical solutions. (C) 2021 Elsevier B.V. All rights reserved.

This paper proposes and analyzes the residual-type a posteriori error estimates for a non-consistent virtual element method (VEM) for reaction–diffusion equations. The reliability and efficiency of the a posteriori error bound are derived under a weak mesh assumption. The error estimators enable us to devise an adaptive VEM by means of the mesh refinement strategy with the one-hanging-node rule. Numerical experiments are in agreement with the theoretical results.

In this paper, three kinds of two-grid Arrow-Hurwicz (A-H) methods are proposed and analyzed for the steady incompressible Navier-Stokes equations, which adopt the existing A-H method to obtain the coarse mesh solution, and further enhance the efficiency by three different one-step schemes (Oseen type, Simple type and Newton type) on the fine mesh. These methods combine the A-H method and the two-grid strategy, retaining the best features of two techniques and overcoming some of their limitations. Furthermore, the error analyses of the three methods are carefully studied and the numerical tests are reported to demonstrate the theoretical results and show the efficiency of the methods.

This paper presents an optimal design of QAM-CCK modulation based on maximizing mapping diversity for ARQ transmission scheme, aiming at improving the unsatisfying performance of QAM-CCK in underwater channels with single transmission. In order to achieve the optimal BER performance, this paper first derives the pair-wise error probability after re-transmission and then obtains the optimal construction of QAM-CCK by minimizing the target function. Simulation results demonstrate that optimum desigend QAM-CCK achieves 2 dB SNR gain compared to randomly selected QAM-CCK when BER equals 10 -4 . Furthermore, optimum desigend QAM-CCK achieves 5 dB SNR gain compared to randomly selected QAMCCK when BER equals 10 -5 under tested acoustic channels from UNet06 sea experiments.

A two-level additive Schwarz preconditioner based on the overlapping domain decomposition approach is proposed for the local $$C^0$$ discontinuous Galerkin (LCDG) method of Kirchhoff plates. Then with the help of an intergrid transfer operator and its error estimates, it is proved that the condition number is bounded by $$O(1+(H^4/\delta ^4))$$ , where H is the diameter of the subdomains and $$\delta $$ measures the overlap among subdomains. And for some special cases of small overlap, the estimate can be improved as $$O(1+(H^3/\delta ^3))$$ . At last, some numerical results are reported to demonstrate the high efficiency of the two-level additive Schwarz preconditioner.

Compressive sensing (CS) aims at decreasing sampling rate to reduce the needed number of samples. On the other hand, the one-bit CS is proposed to reduce the quantization bit. In this paper, we proposed a one-bit CS system aiming at acquiring the multiband sparse signal which is a very popular signal model in wireless communication, especially in cognitive radio. This proposed system, called direct one-bit sampler (DOS), is simple in hardware implementation, and it consists of only a comparator working at Nyquist rate. In the stage of signal reconstruction, it can be equivalent to a special multicoset sampler which is a popular scheme in CS. Moreover, we propose an enhanced binary iterative hard thresholding (BIHT), a popular one-bit recovery algorithm, to deal with the multiple measurement vectors in the one-bit CS framework. Both the theoretical model and experimental results demonstrate that the proposed DOS, with the help of the enhanced BIHT, can not only accurately recover the positions of active subbands of the multiband sparse signal but also roughly estimate the power of each active subband.

This paper is on the numerical solution of an elliptic hemivariational inequality by the virtual element method. We introduce an abstract framework of the numerical method and provide an error analysis. We then apply the virtual element method to solve two contact problems: a bilateral contact problem with friction and a frictionless normal compliance contact problem. Error estimates of their numerical solutions are derived, which are of optimal order for the linear virtual element method, under appropriate solution regularity assumptions. The discrete problem can be formulated as an optimization problem for a difference of two convex (DC) functions, and a convergent algorithm is introduced to solve it. Numerical examples are reported to show the performance of the proposed methods.

With the introduction of numerical traces respectively related to the normal bending moment, the twisting moment and the effective transverse shear force, and based on the Hermann–Miyoshi formulation, this paper proposes a hybridizable discontinuous Galerkin (HDG) method for Kirchhoff plate bending problems. The piecewise polynomials of degrees \(k-1\) and k are used to approximate the moment and the deflection, respectively. The optimal and superconvergent error estimates are derived under minimal regularity assumptions on the exact solution. The key ingredients in the analysis include the derivation of a discrete inf-sup condition and some local lower bound estimates of a posteriori error analysis. The significant feature of the HDG method is superconvergence as well as the low number of globally coupled degrees of freedom associated with Lagrange multipliers. Furthermore, a new discrete deflection is constructed by postprocessing the solution of the HDG method, which superconverges to the deflection with order \(k+1\) in broken \(H^1\) norm. Finally, some numerical results are shown to demonstrate the theoretical results.

Driven by the huge demand to explore oceans, underwater wireless communications have been rapidly developed in the past few decades. Due to the complex physical characteristics of water, acoustic wave is the only media available for underwater wireless communication at any distance. As a result, underwater acoustic communication (UAC) is the major research field in underwater wireless communication. In this paper, characteristics of underwater acoustic channels are first introduced and compared with terrestrial communication to demonstrate the difficulties in UAC research. To give a general impression of the UAC, current important research areas are mentioned. Furthermore, different principal modulation-based schemes for short- and medium-range communications with high data rates are investigated and summarized. To evaluate the performance of UAC systems in general, three criteria are presented based on the research publications and our years of experience in high-rate short- to medium-range communications. These three criteria provide useful tools to generally guide the design and evaluate the performance of underwater acoustic communication systems.