Search

This paper is concerned with developing and analyzing two novel implicit temporal discretization methods for the stochastic semilinear wave equations with multiplicative noise. The proposed methods are natural extensions of well-known time-discrete schemes for deterministic wave equations, hence, they are easy to implement. It is proved that both methods are energy-stable. Moreover, the first method is shown to converge with the linear order in the energy norm, while the second method converges with the $\mathcal{O}(\tau^{\frac32})$ order in the $L^2$-norm, which is optimal with respect to the time regularity of the solution to the underlying stochastic PDE. The convergence analyses of both methods, which are different and quite involved, require some novel numerical techniques to overcome difficulties caused by the nonlinear noise term and the interplay between nonlinear drift and diffusion. Numerical experiments are provided to validate the sharpness of the theoretical error estimate results., Comment: 28 pages, 0 figure, 3 tables

We present DeepSeek-Coder-V2, an open-source Mixture-of-Experts (MoE) code language model that achieves performance comparable to GPT4-Turbo in code-specific tasks. Specifically, DeepSeek-Coder-V2 is further pre-trained from an intermediate checkpoint of DeepSeek-V2 with additional 6 trillion tokens. Through this continued pre-training, DeepSeek-Coder-V2 substantially enhances the coding and mathematical reasoning capabilities of DeepSeek-V2, while maintaining comparable performance in general language tasks. Compared to DeepSeek-Coder-33B, DeepSeek-Coder-V2 demonstrates significant advancements in various aspects of code-related tasks, as well as reasoning and general capabilities. Additionally, DeepSeek-Coder-V2 expands its support for programming languages from 86 to 338, while extending the context length from 16K to 128K. In standard benchmark evaluations, DeepSeek-Coder-V2 achieves superior performance compared to closed-source models such as GPT4-Turbo, Claude 3 Opus, and Gemini 1.5 Pro in coding and math benchmarks.

We introduce a new concept of the locally conservative flux and investigate its relationship with the compatible discretization pioneered by Dawson, Sun and Wheeler [11]. We then demonstrate how the new concept of the locally conservative flux can play a crucial role in obtaining the L2 norm stability of the discontinuous Galerkin finite element scheme for the transport in the coupled system with flow. In particular, the lowest order discontinuous Galerkin finite element for the transport is shown to inherit the positivity and maximum principle when the locally conservative flux is used, which has been elusive for many years in literature. The theoretical results established in this paper are based on the equivalence between Lesaint-Raviart discontinuous Galerkin scheme and Brezzi-Marini-Suli discontinuous Galerkin scheme for the linear hyperbolic system as well as the relationship between the Lesaint-Raviart discontinuous Galerkin scheme and the characteristic method along the streamline. Sample numerical experiments have also been performed to justify our theoretical findings

Diffusion models have been popular for point cloud generation tasks. Existing works utilize the forward diffusion process to convert the original point distribution into a noise distribution and then learn the reverse diffusion process to recover the point distribution from the noise distribution. However, the reverse diffusion process can produce samples with non-smooth points on the surface because of the ignorance of the point cloud geometric properties. We propose alleviating the problem by incorporating the local smoothness constraint into the diffusion framework for point cloud generation. Experiments demonstrate the proposed model can generate realistic shapes and smoother point clouds, outperforming multiple state-of-the-art methods.

This paper explores the problem of continual learning (CL) of vision-language models (VLMs) in open domains, where the models need to perform continual updating and inference on a streaming of datasets from diverse seen and unseen domains with novel classes. Such a capability is crucial for various applications in open environments, e.g., AI assistants, autonomous driving systems, and robotics. Current CL studies mostly focus on closed-set scenarios in a single domain with known classes. Large pre-trained VLMs like CLIP have demonstrated superior zero-shot recognition ability, and a number of recent studies leverage this ability to mitigate catastrophic forgetting in CL, but they focus on closed-set CL in a single domain dataset. Open-domain CL of large VLMs is significantly more challenging due to 1) large class correlations and domain gaps across the datasets and 2) the forgetting of zero-shot knowledge in the pre-trained VLMs in addition to the knowledge learned from the newly adapted datasets. In this work we introduce a novel approach, termed CoLeCLIP, that learns an open-domain CL model based on CLIP. It addresses these challenges by a joint learning of a set of task prompts and a cross-domain class vocabulary. Extensive experiments on 11 domain datasets show that CoLeCLIP outperforms state-of-the-art methods for open-domain CL under both task- and class-incremental learning settings.

This paper proposes and analyzes a novel fully discrete finite element scheme with the interpolation operator for stochastic Cahn-Hilliard equations with functional-type noise. The nonlinear term satisfies a one-side Lipschitz condition and the diffusion term is globally Lipschitz continuous. The novelties of this paper are threefold. First, the $L^2$-stability ($L^\infty$ in time) and the discrete $H^2$-stability ($L^2$ in time) are proved for the proposed scheme. The idea is to utilize the special structure of the matrix assembled by the nonlinear term. None of these stability results has been proved for the fully implicit scheme in existing literature due to the difficulty arising from the interaction of the nonlinearity and the multiplicative noise. Second, the higher moment stability in $L^2$-norm of the discrete solution is established based on the previous stability results. Third, the H\"older continuity in time for the strong solution is established under the minimum assumption of the strong solution. Based on these, the discrete $H^{-1}$-norm of the strong convergence is discussed. Several numerical experiments including stability and convergence are also presented to validate our theoretical results., Comment: 8 figures, 1 table

In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the diffusion term is assumed to be globally Lipschitz continuous. Our new strategy for time discretization is based on the Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly $1$ for the approximate solution. The proof is based on new H\"older continuity estimates of the SPDE solution and the nonlinear term. For the general polynomial-type drift term, there are difficulties in deriving even the stability of the numerical solutions. We propose an interpolation-based finite element method for spatial discretization to overcome the difficulties. Then we obtain $H^1$ stability, higher moment $H^1$ stability, $L^2$ stability, and higher moment $L^2$ stability results using numerical and stochastic techniques. The nearly optimal convergence orders in time and space are hence obtained by coupling all previous results. Numerical experiments are presented to implement the proposed numerical scheme and to validate the theoretical results., Comment: 28 pages, 8 figures. arXiv admin note: text overlap with arXiv:1811.05028

Deep neural networks (DNNs) have achieved tremendous success in making accurate predictions for computer vision, natural language processing, as well as science and engineering domains. However, it is also well-recognized that DNNs sometimes make unexpected, incorrect, but overconfident predictions. This can cause serious consequences in high-stake applications, such as autonomous driving, medical diagnosis, and disaster response. Uncertainty quantification (UQ) aims to estimate the confidence of DNN predictions beyond prediction accuracy. In recent years, many UQ methods have been developed for DNNs. It is of great practical value to systematically categorize these UQ methods and compare their advantages and disadvantages. However, existing surveys mostly focus on categorizing UQ methodologies from a neural network architecture perspective or a Bayesian perspective and ignore the source of uncertainty that each methodology can incorporate, making it difficult to select an appropriate UQ method in practice. To fill the gap, this paper presents a systematic taxonomy of UQ methods for DNNs based on the types of uncertainty sources (data uncertainty versus model uncertainty). We summarize the advantages and disadvantages of methods in each category. We show how our taxonomy of UQ methodologies can potentially help guide the choice of UQ method in different machine learning problems (e.g., active learning, robustness, and reinforcement learning). We also identify current research gaps and propose several future research directions., Comment: 41 pages, 13 figures

Learning to generate graphs is challenging as a graph is a set of pairwise connected, unordered nodes encoding complex combinatorial structures. Recently, several works have proposed graph generative models based on normalizing flows or score-based diffusion models. However, these models need to generate nodes and edges in parallel from the same process, whose dimensionality is unnecessarily high. We propose NVDiff, which takes the VGAE structure and uses a score-based generative model (SGM) as a flexible prior to sample node vectors. By modeling only node vectors in the latent space, NVDiff significantly reduces the dimension of the diffusion process and thus improves sampling speed. Built on the NVDiff framework, we introduce an attention-based score network capable of capturing both local and global contexts of graphs. Experiments indicate that NVDiff significantly reduces computations and can model much larger graphs than competing methods. At the same time, it achieves superior or competitive performances over various datasets compared to previous methods.

This paper is concerned with fully discrete finite element methods for approximating variational solutions of nonlinear stochastic elastic wave equations with multiplicative noise. A detailed analysis of the properties of the weak solution is carried out and a fully discrete finite element method is proposed. Strong convergence in the energy norm with rate $\cal{O}(k+h^r)$ is proved, where $k$ and $h$ denote respectively the temporal and spatial mesh sizes, and $r(\geq 1)$ is the order of the finite element. Numerical experiments are provided to test the efficiency of proposed numerical methods and to validate the theoretical error estimate results., Comment: 37 pages, 4 tables, and 16 graphics

This technical report presents an overview of our solution used in the submission to 2021 HACS Temporal Action Localization Challenge on both Supervised Learning Track and Weakly-Supervised Learning Track. Temporal Action Localization (TAL) requires to not only precisely locate the temporal boundaries of action instances, but also accurately classify the untrimmed videos into specific categories. However, Weakly-Supervised TAL indicates locating the action instances using only video-level class labels. In this paper, to train a supervised temporal action localizer, we adopt Temporal Context Aggregation Network (TCANet) to generate high-quality action proposals through ``local and global" temporal context aggregation and complementary as well as progressive boundary refinement. As for the WSTAL, a novel framework is proposed to handle the poor quality of CAS generated by simple classification network, which can only focus on local discriminative parts, rather than locate the entire interval of target actions. Further inspired by the transfer learning method, we also adopt an additional module to transfer the knowledge from trimmed videos (HACS Clips dataset) to untrimmed videos (HACS Segments dataset), aiming at promoting the classification performance on untrimmed videos. Finally, we employ a boundary regression module embedded with Outer-Inner-Contrastive (OIC) loss to automatically predict the boundaries based on the enhanced CAS. Our proposed scheme achieves 39.91 and 29.78 average mAP on the challenge testing set of supervised and weakly-supervised temporal action localization track respectively., Comment: Winner of HACS21 Challenge Weakly Supervised Learning Track with extra data. arXiv admin note: text overlap with arXiv:2103.13141

Wave propagation problems have many applications in physics and engineering, and the stochastic effects are important in accurately modeling them due to the uncertainty of the media. This paper considers and analyzes a fully discrete finite element method for a class of nonlinear stochastic wave equations, where the diffusion term is globally Lipschitz continuous while the drift term is only assumed to satisfy weaker conditions as in [11]. The novelties of this paper are threefold. First, the error estimates cannot not be directly obtained if the numerical scheme in primal form is used. The numerical scheme in mixed form is introduced and several H\"{o}lder continuity results of the strong solution are proved, which are used to establish the error estimates in both $L^2$ norm and energy norms. Second, two types of discretization of the nonlinear term are proposed to establish the $L^2$ stability and energy stability results of the discrete solutions. These two types of discretization and proper test functions are designed to overcome the challenges arising from the stochastic scaling in time issues and the nonlinear interaction. These stability results play key roles in proving the probability of the set on which the error estimates hold approaches one. Third, higher order moment stability results of the discrete solutions are proved based on an energy argument and the underlying energy decaying property of the method. Numerical experiments are also presented to show the stability results of the discrete solutions and the convergence rates in various norms., Comment: 26 pages, 5 figures

Retrieval is a crucial stage in web search that identifies a small set of query-relevant candidates from a billion-scale corpus. Discovering more semantically-related candidates in the retrieval stage is very promising to expose more high-quality results to the end users. However, it still remains non-trivial challenges of building and deploying effective retrieval models for semantic matching in real search engine. In this paper, we describe the retrieval system that we developed and deployed in Baidu Search. The system exploits the recent state-of-the-art Chinese pretrained language model, namely Enhanced Representation through kNowledge IntEgration (ERNIE), which facilitates the system with expressive semantic matching. In particular, we developed an ERNIE-based retrieval model, which is equipped with 1) expressive Transformer-based semantic encoders, and 2) a comprehensive multi-stage training paradigm. More importantly, we present a practical system workflow for deploying the model in web-scale retrieval. Eventually, the system is fully deployed into production, where rigorous offline and online experiments were conducted. The results show that the system can perform high-quality candidate retrieval, especially for those tail queries with uncommon demands. Overall, the new retrieval system facilitated by pretrained language model (i.e., ERNIE) can largely improve the usability and applicability of our search engine., Comment: Accepted by KDD 2021