The existence of smooth but nowhere analytic functions is well-known (du Bois-Reymond, Math. Ann., 21(1):109-117, 1883). However, smooth solutions to the heat equation are usually analytic in the space variable. It is also well-known (Kowalevsky, Cre
lle, 80:1-32, 1875) that a solution to the heat equation may not be time-analytic at $t=0$ even if the initial function is real analytic. Recently, it was shown in cite{Zha20, DZ20, DP20} that solutions to the heat equation in the whole space, or half space with zero boundary value, are analytic in time under essentially optimal conditions. In this paper, we show that time analyticity is not always true in domains with general boundary conditions or without suitable growth conditions. More precisely, we construct two bounded solutions to the heat equation in the half plane which are nowhere analytic in time. In addition, for any $delta>0$, we find a solution to the heat equation on the whole plane, with exponential growth of order $2+delta$, which is nowhere analytic in time.
Deep learning and data-driven approaches have shown great potential in scientific domains. The promise of data-driven techniques relies on the availability of a large volume of high-quality training datasets. Due to the high cost of obtaining data th
rough expensive physical experiments, instruments, and simulations, data augmentation techniques for scientific applications have emerged as a new direction for obtaining scientific data recently. However, existing data augmentation techniques originating from computer vision, yield physically unacceptable data samples that are not helpful for the domain problems that we are interested in. In this paper, we develop new physics-informed data augmentation techniques based on convolutional neural networks. Specifically, our generative models leverage different physics knowledge (such as governing equations, observable perception, and physics phenomena) to improve the quality of the synthetic data. To validate the effectiveness of our data augmentation techniques, we apply them to solve a subsurface seismic full-waveform inversion using simulated CO$_2$ leakage data. Our interest is to invert for subsurface velocity models associated with very small CO$_2$ leakage. We validate the performance of our methods using comprehensive numerical tests. Via comparison and analysis, we show that data-driven seismic imaging can be significantly enhanced by using our physics-informed data augmentation techniques. Particularly, the imaging quality has been improved by 15% in test scenarios of general-sized leakage and 17% in small-sized leakage when using an augmented training set obtained with our techniques.
This paper investigates how to realize better and more efficient embedding learning to tackle the semi-supervised video object segmentation under challenging multi-object scenarios. The state-of-the-art methods learn to decode features with a single
positive object and thus have to match and segment each target separately under multi-object scenarios, consuming multiple times computing resources. To solve the problem, we propose an Associating Objects with Transformers (AOT) approach to match and decode multiple objects uniformly. In detail, AOT employs an identification mechanism to associate multiple targets into the same high-dimensional embedding space. Thus, we can simultaneously process the matching and segmentation decoding of multiple objects as efficiently as processing a single object. For sufficiently modeling multi-object association, a Long Short-Term Transformer is designed for constructing hierarchical matching and propagation. We conduct extensive experiments on both multi-object and single-object benchmarks to examine AOT variant networks with different complexities. Particularly, our AOT-L outperforms all the state-of-the-art competitors on three popular benchmarks, i.e., YouTube-VOS (83.7% J&F), DAVIS 2017 (83.0%), and DAVIS 2016 (91.0%), while keeping more than 3X faster multi-object run-time. Meanwhile, our AOT-T can maintain real-time multi-object speed on the above benchmarks. We ranked 1st in the 3rd Large-scale Video Object Segmentation Challenge. The code will be publicly available at https://github.com/z-x-yang/AOT.
Simultaneous translation, which starts translating each sentence after receiving only a few words in source sentence, has a vital role in many scenarios. Although the previous prefix-to-prefix framework is considered suitable for simultaneous transla
tion and achieves good performance, it still has two inevitable drawbacks: the high computational resource costs caused by the need to train a separate model for each latency $k$ and the insufficient ability to encode information because each target token can only attend to a specific source prefix. We propose a novel framework that adopts a simple but effective decoding strategy which is designed for full-sentence models. Within this framework, training a single full-sentence model can achieve arbitrary given latency and save computational resources. Besides, with the competence of the full-sentence model to encode the whole sentence, our decoding strategy can enhance the information maintained in the decoded states in real time. Experimental results show that our method achieves better translation quality than baselines on 4 directions: Zh$rightarrow$En, En$rightarrow$Ro and En$leftrightarrow$De.
Image matting is an ill-posed problem that usually requires additional user input, such as trimaps or scribbles. Drawing a fne trimap requires a large amount of user effort, while using scribbles can hardly obtain satisfactory alpha mattes for non-pr
ofessional users. Some recent deep learning-based matting networks rely on large-scale composite datasets for training to improve performance, resulting in the occasional appearance of obvious artifacts when processing natural images. In this article, we explore the intrinsic relationship between user input and alpha mattes and strike a balance between user effort and the quality of alpha mattes. In particular, we propose an interactive framework, referred to as smart scribbles, to guide users to draw few scribbles on the input images to produce high-quality alpha mattes. It frst infers the most informative regions of an image for drawing scribbles to indicate different categories (foreground, background, or unknown) and then spreads these scribbles (i.e., the category labels) to the rest of the image via our well-designed two-phase propagation. Both neighboring low-level afnities and high-level semantic features are considered during the propagation process. Our method can be optimized without large-scale matting datasets and exhibits more universality in real situations. Extensive experiments demonstrate that smart scribbles can produce more accurate alpha mattes with reduced additional input, compared to the state-of-the-art matting methods.
We develop covariant chiral kinetic theory with Landau level basis. We use it to investigate a magnetized plasma with a transverse electric field and a steady vorticity as perturbations. After taking into account vacuum shift in the latter case, we f
ind the resulting current and stress tensor in both cases can be matched consistently with constitutive equations of magnetohydrodynamics. We find the solution in the vorticity case contains both shifts in temperature and chemical potential as well as excitations of the lowest Landau level states. The solution gives rise to an vector charge density and axial current density. The vacuum parts coming from both shifts and excitations agree with previous studies and the medium parts coming entirely from excitations leads to a new contribution to vector charge and axial current density consistent with standard chiral vortical effect.
Modeling the hand-object (HO) interaction not only requires estimation of the HO pose, but also pays attention to the contact due to their interaction. Significant progress has been made in estimating hand and object separately with deep learning met
hods, simultaneous HO pose estimation and contact modeling has not yet been fully explored. In this paper, we present an explicit contact representation namely Contact Potential Field (CPF), and a learning-fitting hybrid framework namely MIHO to Modeling the Interaction of Hand and Object. In CPF, we treat each contacting HO vertex pair as a spring-mass system. Hence the whole system forms a potential field with minimal elastic energy at the grasp position. Extensive experiments on the two commonly used benchmarks have demonstrated that our method can achieve state-of-the-art in several reconstruction metrics, and allow us to produce more physically plausible HO pose even when the ground-truth exhibits severe interpenetration or disjointedness. Our code is available at https://github.com/lixiny/CPF.
3D ultrasound (US) is widely used due to its rich diagnostic information, portability and low cost. Automated standard plane (SP) localization in US volume not only improves efficiency and reduces user-dependence, but also boosts 3D US interpretation
. In this study, we propose a novel Multi-Agent Reinforcement Learning (MARL) framework to localize multiple uterine SPs in 3D US simultaneously. Our contribution is two-fold. First, we equip the MARL with a one-shot neural architecture search (NAS) module to obtain the optimal agent for each plane. Specifically, Gradient-based search using Differentiable Architecture Sampler (GDAS) is employed to accelerate and stabilize the training process. Second, we propose a novel collaborative strategy to strengthen agents communication. Our strategy uses recurrent neural network (RNN) to learn the spatial relationship among SPs effectively. Extensively validated on a large dataset, our approach achieves the accuracy of 7.05 degree/2.21mm, 8.62 degree/2.36mm and 5.93 degree/0.89mm for the mid-sagittal, transverse and coronal plane localization, respectively. The proposed MARL framework can significantly increase the plane localization accuracy and reduce the computational cost and model size.
Magnetocrystalline anisotropy is a fundamental property of magnetic materials that determines the dynamics of magnetic precession, the frequency of spin waves, the thermal stability of magnetic domains, and the efficiency of spintronic devices. We co
mbine torque magnetometry and density functional theory calculations to determine the magnetocrystalline anisotropy of the metallic antiferromagnet Fe$_2$As. Fe$_2$As has a tetragonal crystal structure with the Neel vector lying in the (001) plane. We report that the four-fold magnetocrystalline anisotropy in the (001)-plane of Fe$_2$As is extremely small, ${K_{22}} = - 150~{rm{ J/}}{{rm{m}}^{rm{3}}}$ at T = 4 K, much smaller than perpendicular magnetic anisotropy of ferromagnetic structure widely used in spintronics device. ${K_{22}}$ is strongly temperature dependent and close to zero at T > 150 K. The anisotropy ${K_1}$ in the (010) plane is too large to be measured by torque magnetometry and we determine ${K_1} = -830~{rm{ kJ/}}{{rm{m}}^{rm{3}}}$ using first-principles density functional theory. Our simulations show that the contribution to the anisotropy from classical magnetic dipole-dipole interactions is comparable to the contribution from spin-orbit coupling. The calculated four-fold anisotropy in the (001) plane ${K_{22}}$ ranges from $- 292~{rm{ J/}}{{rm{m}}^{rm{3}}}$ to $280~{rm{ J/}}{{rm{m}}^{rm{3}}}$, the same order of magnitude as the measured value. We use ${K_1}$ from theory to predict the frequency and polarization of the lowest frequency antiferromagnetic resonance mode and find that the mode is linearly polarized in the (001)-plane with $f = $ 670 GHz.
Background and Objective: Biometric measurements of fetal head are important indicators for maternal and fetal health monitoring during pregnancy. 3D ultrasound (US) has unique advantages over 2D scan in covering the whole fetal head and may promote
the diagnoses. However, automatically segmenting the whole fetal head in US volumes still pends as an emerging and unsolved problem. The challenges that automated solutions need to tackle include the poor image quality, boundary ambiguity, long-span occlusion, and the appearance variability across different fetal poses and gestational ages. In this paper, we propose the first fully-automated solution to segment the whole fetal head in US volumes. Methods: The segmentation task is firstly formulated as an end-to-end volumetric mapping under an encoder-decoder deep architecture. We then combine the segmentor with a proposed hybrid attention scheme (HAS) to select discriminative features and suppress the non-informative volumetric features in a composite and hierarchical way. With little computation overhead, HAS proves to be effective in addressing boundary ambiguity and deficiency. To enhance the spatial consistency in segmentation, we further organize multiple segmentors in a cascaded fashion to refine the results by revisiting context in the prediction of predecessors. Results: Validated on a large dataset collected from 100 healthy volunteers, our method presents superior segmentation performance (DSC (Dice Similarity Coefficient), 96.05%), remarkable agreements with experts. With another 156 volumes collected from 52 volunteers, we ahieve high reproducibilities (mean standard deviation 11.524 mL) against scan variations. Conclusion: This is the first investigation about whole fetal head segmentation in 3D US. Our method is promising to be a feasible solution in assisting the volumetric US-based prenatal studies.
