Search

In this paper, we examine a fully-discrete finite element approximation of the unsteady $p(\cdot,\cdot)$-Stokes equations ($i.e.$, $p(\cdot,\cdot)$ is time- and space-dependent), employing a backward Euler step in time and conforming, discretely inf-sup stable finite elements in space. More precisely, we derive error decay rates for the vector-valued velocity field imposing fractional regularity assumptions on the velocity and the kinematic pressure. In addition, we carry out numerical experiments that confirm the optimality of the derived error decay rates in the case $p(\cdot,\cdot)\ge 2$., Comment: 27 pages, 4 tables

Exploration is a critical challenge in robotics, centered on understanding unknown environments. In this work, we focus on robots exploring structured indoor environments which are often predictable and composed of repeating patterns. Most existing approaches, such as conventional frontier approaches, have difficulty leveraging the predictability and explore with simple heuristics such as `closest first'. Recent works use deep learning techniques to predict unknown regions of the map, using these predictions for information gain calculation. However, these approaches are often sensitive to the predicted map quality or do not reason over sensor coverage. To overcome these issues, our key insight is to jointly reason over what the robot can observe and its uncertainty to calculate probabilistic information gain. We introduce MapEx, a new exploration framework that uses predicted maps to form probabilistic sensor model for information gain estimation. MapEx generates multiple predicted maps based on observed information, and takes into consideration both the computed variances of predicted maps and estimated visible area to estimate the information gain of a given viewpoint. Experiments on the real-world KTH dataset showed on average 12.4% improvement than representative map-prediction based exploration and 25.4% improvement than nearest frontier approach., Comment: 7 pages

Entanglement is a fundamental aspect of quantum physics, both conceptually and for its many applications. Classifying an arbitrary multipartite state as entangled or separable -- a task referred to as the separability problem -- poses a significant challenge, since a state can be entangled with respect to many different of its partitions. We develop a certification pipeline that feeds the statistics of random local measurements into a non-linear dimensionality reduction algorithm, to determine with respect to which partitions a given quantum state is entangled. After training a model on randomly generated quantum states, entangled in different partitions and of varying purity, we verify the accuracy of its predictions on simulated test data, and finally apply it to states prepared on IBM quantum computing hardware., Comment: 15 pages, 5 figures

We consider a system of nonlinear partial differential equations modeling the unsteady motion of an incompressible generalized Newtonian fluid with chemical reactions. The system consists of the generalized Navier-Stokes equations with power-law type viscosity with a power-law index depending on the concentration, and the convection-diffusion equation which describes chemical concentration. This system of partial differential equations arises in the mathematical models describing the synovial fluid which can be found in the cavities of movable joints. We prove the existence of a global strong solution for the two and three-dimensional spatially periodic domain, provided that the power-law index is greater than or equal to $(d+2)/2$ where $d$ is the dimension of the spatial domain. Moreover, we also prove that such a solution is unique under the further assumption that $p^+ < \frac{3}{2} p^-$ for the two-dimensional case and $p^+ < \frac{7}{6}p^-$ for the three-dimensional case, where $p^-$ and $p^+$ are the lower and upper bounds of the power-law index $p(\cdot)$ respectively.

Physics-informed neural networks (PINNs) have recently emerged as a promising way to compute the solutions of partial differential equations (PDEs) using deep neural networks. However, despite their significant success in various fields, it remains unclear in many aspects how to effectively train PINNs if the solutions of PDEs exhibit stiff behaviors or high frequencies. In this paper, we propose a new method for training PINNs using variable-scaling techniques. This method is simple and it can be applied to a wide range of problems including PDEs with rapidly-varying solutions. Throughout various numerical experiments, we will demonstrate the effectiveness of the proposed method for these problems and confirm that it can significantly improve the training efficiency and performance of PINNs. Furthermore, based on the analysis of the neural tangent kernel (NTK), we will provide theoretical evidence for this phenomenon and show that our methods can indeed improve the performance of PINNs.

State preparation that initializes quantum systems in a fiducial state and measurements to read outcomes after the evolution of quantum states, both essential elements in quantum information processing in general, may contain noise from which errors, in particular, referred to as static errors, may appear even with noise-free evolution. In this work, we consider noisy resources such as faulty manufacturing and improper maintenance of systems by which static errors in state preparation and measurement are inevitable. We show how to suppress static errors and purify noiseless SPAM by repeatedly applying noisy resources. We present the purification protocol for noisy initialization and noisy measurements and verify that a few qubits are immediately cost-effective to suppress error rates up to $10^{-3}$. We also demonstrate the purification protocol in a realistic scenario. The results are readily feasible with current quantum technologies., Comment: 9 pages, 5 figures

In this paper, we provide a theoretical analysis of a type of operator learning method without data reliance based on the classical finite element approximation, which is called the finite element operator network (FEONet). We first establish the convergence of this method for general second-order linear elliptic PDEs with respect to the parameters for neural network approximation. In this regard, we address the role of the condition number of the finite element matrix in the convergence of the method. Secondly, we derive an explicit error estimate for the self-adjoint case. For this, we investigate some regularity properties of the solution in certain function classes for a neural network approximation, verifying the sufficient condition for the solution to have the desired regularity. Finally, we will also conduct some numerical experiments that support the theoretical findings, confirming the role of the condition number of the finite element matrix in the overall convergence.

This paper presents a novel method for reconstructing 3D garment models from a single image of a posed user. Previous studies that have primarily focused on accurately reconstructing garment geometries to match the input garment image may often result in unnatural-looking garments when deformed for new poses. To overcome this limitation, our approach takes a different approach by inferring the fundamental shape of the garment through sewing patterns from a single image, rather than directly reconstructing 3D garments. Our method consists of two stages. Firstly, given a single image of a posed user, it predicts the garment image worn on a T-pose, representing the baseline form of the garment. Then, it estimates the sewing pattern parameters based on the T-pose garment image. By simulating the stitching and draping of the sewing pattern using physics simulation, we can generate 3D garments that can adaptively deform to arbitrary poses. The effectiveness of our method is validated through ablation studies on the major components and a comparison with other approaches.

Building general-purpose robots that operate seamlessly in any environment, with any object, and utilizing various skills to complete diverse tasks has been a long-standing goal in Artificial Intelligence. However, as a community, we have been constraining most robotic systems by designing them for specific tasks, training them on specific datasets, and deploying them within specific environments. These systems require extensively-labeled data and task-specific models. When deployed in real-world scenarios, such systems face several generalization issues and struggle to remain robust to distribution shifts. Motivated by the impressive open-set performance and content generation capabilities of web-scale, large-capacity pre-trained models (i.e., foundation models) in research fields such as Natural Language Processing (NLP) and Computer Vision (CV), we devote this survey to exploring (i) how these existing foundation models from NLP and CV can be applied to the field of general-purpose robotics, and also exploring (ii) what a robotics-specific foundation model would look like. We begin by providing a generalized formulation of how foundation models are used in robotics, and the fundamental barriers to making generalist robots universally applicable. Next, we establish a taxonomy to discuss current work exploring ways to leverage existing foundation models for robotics and develop ones catered to robotics. Finally, we discuss key challenges and promising future directions in using foundation models for enabling general-purpose robotic systems. We encourage readers to view our living GitHub repository 2 of resources, including papers reviewed in this survey, as well as related projects and repositories for developing foundation models for robotics.

Partial differential equations (PDEs) underlie our understanding and prediction of natural phenomena across numerous fields, including physics, engineering, and finance. However, solving parametric PDEs is a complex task that necessitates efficient numerical methods. In this paper, we propose a novel approach for solving parametric PDEs using a Finite Element Operator Network (FEONet). Our proposed method leverages the power of deep learning in conjunction with traditional numerical methods, specifically the finite element method, to solve parametric PDEs in the absence of any paired input-output training data. We performed various experiments on several benchmark problems and confirmed that our approach has demonstrated excellent performance across various settings and environments, proving its versatility in terms of accuracy, generalization, and computational flexibility. Our FEONet framework shows potential for application in various fields where PDEs play a crucial role in modeling complex domains with diverse boundary conditions and singular behavior. Furthermore, we provide theoretical convergence analysis to support our approach, utilizing finite element approximation in numerical analysis., Comment: 23 pages, 11 figures

This work proposes an autonomous multi-robot exploration pipeline that coordinates the behaviors of robots in an indoor environment composed of multiple rooms. Contrary to simple frontier-based exploration approaches, we aim to enable robots to methodically explore and observe an unknown set of rooms in a structured building, keeping track of which rooms are already explored and sharing this information among robots to coordinate their behaviors in a distributed manner. To this end, we propose (1) a geometric cue extraction method that processes 3D point cloud data and detects the locations of potential cues such as doors and rooms, (2) a circular decomposition for free spaces used for target assignment. Using these two components, our pipeline effectively assigns tasks among robots, and enables a methodical exploration of rooms. We evaluate the performance of our pipeline using a team of up to 3 aerial robots, and show that our method outperforms the baseline by 33.4% in simulation and 26.4% in real-world experiments.

In this paper, we perform the convergence analysis of unsupervised Legendre--Galerkin neural networks (ULGNet), a deep-learning-based numerical method for solving partial differential equations (PDEs). Unlike existing deep learning-based numerical methods for PDEs, the ULGNet expresses the solution as a spectral expansion with respect to the Legendre basis and predicts the coefficients with deep neural networks by solving a variational residual minimization problem. Since the corresponding loss function is equivalent to the residual induced by the linear algebraic system depending on the choice of basis functions, we prove that the minimizer of the discrete loss function converges to the weak solution of the PDEs. Numerical evidence will also be provided to support the theoretical result. Key technical tools include the variant of the universal approximation theorem for bounded neural networks, the analysis of the stiffness and mass matrices, and the uniform law of large numbers in terms of the Rademacher complexity.

In this paper, we analyze the numerical approximation of the Navier-Stokes problem over a bounded polygonal domain in $\mathbb{R}^2$, where the initial condition is modeled by a log-normal random field. This problem usually arises in the area of uncertainty quantification. We aim to compute the expectation value of linear functionals of the solution to the Navier-Stokes equations and perform a rigorous error analysis for the problem. In particular, our method includes the finite element, fully-discrete discretizations, truncated Karhunen-Lo\'eve expansion for the realizations of the initial condition, and lattice-based quasi-Monte Carlo (QMC) method to estimate the expected values over the parameter space. Our QMC analysis is based on randomly-shifted lattice rules for the integration over the domain in high-dimensional space, which guarantees the error decays with $\mathcal{O}(N^{-1+\delta})$, where $N$ is the number of sampling points, $\delta>0$ is an arbitrary small number, and the constant in the decay estimate is independent of the dimension of integration.

In semiconductor manufacturing, wafer map defect pattern provides critical information for facility maintenance and yield management, so the classification of defect patterns is one of the most important tasks in the manufacturing process. In this paper, we propose a novel way to represent the shape of the defect pattern as a finite-dimensional vector, which will be used as an input for a neural network algorithm for classification. The main idea is to extract the topological features of each pattern by using the theory of persistent homology from topological data analysis (TDA). Through some experiments with a simulated dataset, we show that the proposed method is faster and much more efficient in training with higher accuracy, compared with the method using convolutional neural networks (CNN) which is the most common approach for wafer map defect pattern classification. Moreover, our method outperforms the CNN-based method when the number of training data is not enough and is imbalanced.

Interestingness recognition is crucial for decision making in autonomous exploration for mobile robots. Previous methods proposed an unsupervised online learning approach that can adapt to environments and detect interesting scenes quickly, but lack the ability to adapt to human-informed interesting objects. To solve this problem, we introduce a human-interactive framework, AirInteraction, that can detect human-informed objects via few-shot online learning. To reduce the communication bandwidth, we first apply an online unsupervised learning algorithm on the unmanned vehicle for interestingness recognition and then only send the potential interesting scenes to a base-station for human inspection. The human operator is able to draw and provide bounding box annotations for particular interesting objects, which are sent back to the robot to detect similar objects via few-shot learning. Only using few human-labeled examples, the robot can learn novel interesting object categories during the mission and detect interesting scenes that contain the objects. We evaluate our method on various interesting scene recognition datasets. To the best of our knowledge, it is the first human-informed few-shot object detection framework for autonomous exploration.

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) causes coronavirus disease 2019 (COVID-19). Imaging tests such as chest X-ray (CXR) and computed tomography (CT) can provide useful information to clinical staff for facilitating a diagnosis of COVID-19 in a more efficient and comprehensive manner. As a breakthrough of artificial intelligence (AI), deep learning has been applied to perform COVID-19 infection region segmentation and disease classification by analyzing CXR and CT data. However, prediction uncertainty of deep learning models for these tasks, which is very important to safety-critical applications like medical image processing, has not been comprehensively investigated. In this work, we propose a novel ensemble deep learning model through integrating bagging deep learning and model calibration to not only enhance segmentation performance, but also reduce prediction uncertainty. The proposed method has been validated on a large dataset that is associated with CXR image segmentation. Experimental results demonstrate that the proposed method can improve the segmentation performance, as well as decrease prediction uncertainties.

In this paper we prove some Liouville-type theorems for the stationary magneto-micropolar fluids under suitable conditions in three space dimensions. We first prove that the solutions are trivial under the assumption of certain growth conditions for the mean oscillations of the potentials. And then we show similar results assuming that the the solutions are contained in $L^p(\mathbb{R}^3)$ with $p\in[2,9/2)$. Finally we show the same result for lower values of $p\in[1,9/4)$ with the further assumption that the solutions vanish at infinity.

We consider the motion of an incompressible shear-thickening power-law-like non-Newtonian fluid in $R^3$ with a variable power-law index. This system of nonlinear partial differential equations arises in mathematical models of electrorheological fluids. The aim of this paper is to investigate the large-time behaviour of the difference $u-\tilde{u}$ where $u$ is a strong solution of the given equations with the initial data $u_0$ and $\tilde{u}$ is the strong solution of the same equations with perturbed initial data $u_0+w_0$. The initial perturbation $w_0$ is not required to be small, but is assumed to satisfy certain decay condition. In particular, we can show that $(1+t)^{-\frac{\gamma}{2}}\lesssim \|u(t)-\tilde{u}(t)\|_2\lesssim(1+t)^{-\frac{\gamma}{2}}$, for sufficiently large $t>0$, where $\gamma\in(2,\frac{5}{2})$. The proof is based on the observation that the solution of the linear heat equation describes the asymptotic behaviour of the solutions of the electrorheological fluids well for sufficiently large time $t>0$, and the generalized Fourier splitting method with an iterative argument. Furthermore, it will also be discussed that the argument used in the present paper can improve the previous results for the generalized Newtonian fluids with a constant power-law index.

We consider the motion of a power-law-like generalized Newtonian fluid in R^3, where the power-law index is a variable function. This system of nonlinear partial differential equations arises in mathematical models of electrorheological fluids. The aim of this paper is to investigate the decay properties of strong solutions for the model, based on the Fourier splitting method. We first prove that the L^2-norm of the solution has the decay rate (1 + t)^{-3/4}. If the H^1-norm of the initial data is sufficiently small, we further show that the derivative of the solution decays in L^2-norm at the rate (1 + t)^{-5/4}.

In topological semimetals and insulators, negative longitudinal magnetoresistance and angle-dependent planar Hall effect have been reported arising from the Berry curvature. Using the Boltzmann transport theory, we present a closed-form expression for the nonequilibrium distribution function which includes both the effects of the Berry curvature and Lorentz force. Using this formulation, we obtain analytical expressions for conductivity and resistivity tensors in Weyl semimetals demonstrating a non-monotonic field dependence arising from the competition between the two effects., Comment: 14 pages, 3 figures

In practical realizations of quantum information processing, there may exist noise in a measurement readout stage where errors appear not only on individual qubits but also on multiple ones collectively, the latter of which is called crosstalk errors. In this work, we present a framework for mitigating measurement errors, for both individual and crosstalk errors. The mitigation protocol consists of two steps, firstly quantum pre-processing, which applies local unitary transformations before a measurement, and classical post-processing that manipulates measurement outcomes to recover noiseless data. The local unitaries in quantum pre-processing can be constructed by characterizing a noisy measurement via quantum detector tomography. We show that the mitigation protocol can maintain a measurement error on multiple qubits as much as that in a single-qubit readout, i.e., the error rates for measurements on multiple qubits are suppressed up to a percent level. The mitigation protocol is realized in IBMQ Sydney and applied to the certification of entanglement-generating circuits. It is demonstrated that the mitigation protocol can successfully eliminate measurement errors so that entanglement-generation circuits can be efficiently certified., Comment: 10 pages, 8 figures, 3 tables

Few-shot object detection has attracted increasing attention and rapidly progressed in recent years. However, the requirement of an exhaustive offline fine-tuning stage in existing methods is time-consuming and significantly hinders their usage in online applications such as autonomous exploration of low-power robots. We find that their major limitation is that the little but valuable information from a few support images is not fully exploited. To solve this problem, we propose a brand new architecture, AirDet, and surprisingly find that, by learning class-agnostic relation with the support images in all modules, including cross-scale object proposal network, shots aggregation module, and localization network, AirDet without fine-tuning achieves comparable or even better results than many fine-tuned methods, reaching up to 30-40% improvements. We also present solid results of onboard tests on real-world exploration data from the DARPA Subterranean Challenge, which strongly validate the feasibility of AirDet in robotics. To the best of our knowledge, AirDet is the first feasible few-shot detection method for autonomous exploration of low-power robots. The code and pre-trained models are released at https://github.com/Jaraxxus-Me/AirDet., Comment: 23 pages, 9 figures

Quantum technologies available currently contain noise in general, often dubbed noisy intermediate-scale quantum (NISQ) systems. We here present the verification of noise in measurement readout errors in cloud-based quantum computing services, IBMQ and Rigetti, by directly performing quantum detector tomography, and show that there exist measurement crosstalk errors. We provide the characterization and the quantification of noise in a quantum measurement of multiple qubits. We remark that entanglement is found as a source of crosstalk errors in a measurement of three qubits., Comment: 8 pages, 2 figures, To be published in IEEE Internet Computing