arXiv subjects
Publications and source records attributed to Jiajun Tong.
For the 2-D incompressible Euler equation, we prove the existence of $m$-fold symmetric uniformly rotating vortex patches with 90-degree corners for all $m\geq 12$. Properties of these patches and the induced stationary flows in the co-rotating frame are characterized. Our construction is based on a novel fixed-point method.
We investigate novel scenarios of self-similar finite-time blowups of the generalized Constantin-Lax-Majda model with a parameter $a$, which are induced by a new setting where the smooth initial data satisfy certain derivative degeneracy condition. In this setting, our numerical study reveals distinct self-similar blowup behaviors depending on the sign of $a$. For $a>0$, we observe one-scale self-similar blowups with regular profiles that have not been found in previous studies. In contrast, for $a\le 0$, we discover a novel two-scale self-similar blowup scenario where the outer profile converges to a singular function at the blowup time while the inner profile remains regular on a much smaller scale. Correspondingly, an $a$-parameterized family of singular self-similar profiles with explicit expressions are constructed for $a<0$ and shown to match nicely with the limiting profiles obtained in numerical simulation. In particular, for the specific case of $a=0$, we rigorously prove the convergence of the outer profile to an explicit singular function in self-similar coordinates. Furthermore, we demonstrate the two-scale nature of the blowup in this scenario by showing that the local inner profile behavior around the singularity point of the outer profile is governed by a traveling wave on a smaller scale. To support this observation, we rigorously establish the existence of such traveling wave solutions via a fixed-point method.
We study the immersed boundary problem in 2-D. It models a 1-D elastic closed string immersed and moving in a fluid that fills the entire plane, where the fluid motion is governed by the 2-D incompressible Navier-Stokes equation with a positive Reynolds number subject to a singular forcing exerted by the string. We introduce the notion of mild solutions to this system, and prove its existence, uniqueness, and optimal regularity estimates when the initial string configuration is $C^1$ and satisfies the well-stretched condition and when the initial flow field $u_0$ lies in $L^p(\mathbb{R}^2)$ with $p\in (2,\infty)$. A blow-up criterion is also established. When the Reynolds number is sent to zero, we show convergence in short time of the solution to that of the Stokes case of 2-D immersed boundary problem, with the optimal error estimates derived. We prove the energy law of the system when $u_0$ additionally belongs to $L^2(\mathbb{R}^2)$. Lastly, we show that the solution is global when the initial data is sufficiently close to an equilibrium state.
We rigorously construct the first steady traveling wave solutions of the 2D incompressible Euler equation that take the form of a contiguous vortex-patch dipole, which can be viewed as the vortex-patch counterpart of the well-known Lamb-Chaplygin dipole. Our construction is based on a novel fixed-point approach that determines the patch boundary as the fixed point of a certain nonlinear map. Smoothness and other properties of the patch boundary are also obtained.
We investigate the general Porous Medium Equations with drift and source terms that model tumor growth. Incompressible limit of such models has been well-studied in the literature, where convergence of the density and pressure variables are established, while it remains unclear whether the free boundaries of the solutions exhibit convergence as well. In this paper, we provide an affirmative result by showing that the free boundaries converge in the Hausdorff distance in the incompressible limit. To achieve this, we quantify the relation between the free boundary motion and spatial average of the pressure, and establish a uniform-in-$m$ strict expansion property of the pressure supports. As a corollary, we derive upper bounds for the Hausdorff dimensions of the free boundaries and show that the limiting free boundary has finite $(d-1)$-dimensional Hausdorff measure.
Commit Classification(CC) is an important task in software maintenance since it helps software developers classify code changes into different types according to their nature and purpose. This allows them to better understand how their development efforts are progressing, identify areas where they need improvement. However, existing methods are all discriminative models, usually with complex architectures that require additional output layers to produce class label probabilities. Moreover, they require a large amount of labeled data for fine-tuning, and it is difficult to learn effective classification boundaries in the case of limited labeled data. To solve above problems, we propose a generative framework that Incorporating prompt-tuning for commit classification with prior knowledge (IPCK) https://github.com/AppleMax1992/IPCK, which simplifies the model structure and learns features across different tasks. It can still reach the SOTA performance with only limited samples. Firstly, we proposed a generative framework based on T5. This encoder-decoder construction method unifies different CC task into a text2text problem, which simplifies the structure of the model by not requiring an extra output layer. Second, instead of fine-tuning, we design an prompt-tuning solution which can be adopted in few-shot scenarios with only limit samples. Furthermore, we incorporate prior knowledge via an external knowledge graph to map the probabilities of words into the final labels in the speech machine step to improve performance in few-shot scenarios. Extensive experiments on two open available datasets show that our framework can solve the CC problem simply but effectively in few-shot and zeroshot scenarios, while improving the adaptability of the model without requiring a large amount of training samples for fine-tuning.
This paper studies mean field games with optimal stopping time (OSMFGs) where agents make optimal exit decisions. Such a model features a coupled obstacle problem and Fokker-Planck equation posing challenges on top of classic mean field games. The nonconvex nature of exit decisions renders the existence of a classic pure strategy equilibrium infeasible, necessitating the consideration of more complex mixed strategy equilibria. This paper proposes a generalized fictitious play algorithm that computes OSMFG mixed equilibria by iteratively solving pure strategy systems, i.e., approximating mixed strategies through averaging pure strategies according to a certain updating rule. The generalized fictitious play allows for a broad family of learning rates and the convergence to the mixed strategy equilibrium can be rigorously justified. The algorithm also incorporates efficient finite difference schemes of the pure strategy system. Numerical experiments demonstrate the effectiveness of the proposed method in robustly and efficiently computing mixed equilibria for OSMFGs.
Commit Classification (CC) is an important task in software maintenance, which helps software developers classify code changes into different types according to their nature and purpose. It allows developers to understand better how their development efforts are progressing, identify areas where they need improvement, and make informed decisions about when and how to release new software versions. However, existing models need lots of manually labeled data for fine-tuning processes, and ignore sentence-level semantic information, which is often essential for discovering the difference between diverse commits. Therefore, it is still challenging to solve CC in fewshot scenario. To solve the above problems, we propose a contrastive learning-based commit classification framework. Firstly, we generate $K$ sentences and pseudo-labels according to the labels of the dataset, which aims to enhance the dataset. Secondly, we randomly group the augmented data $N$ times to compare their similarity with the positive $T_p^{|C|}$ and negative $T_n^{|C|}$ samples. We utilize individual pretrained sentence transformers (ST)s to efficiently obtain the sentence-level embeddings from different features respectively. Finally, we adopt the cosine similarity function to limit the distribution of vectors, similar vectors are more adjacent. The light fine-tuned model is then applied to the label prediction of incoming commits. Extensive experiments on two open available datasets demonstrate that our framework can solve the CC problem simply but effectively in fewshot scenarios, while achieving state-of-the-art(SOTA) performance and improving the adaptability of the model without requiring a large number of training samples for fine-tuning. The code, data, and trained models are available at https://github.com/AppleMax1992/CommitFit.
The 2-D Peskin problem describes a 1-D closed elastic string immersed and moving in a 2-D Stokes flow that is induced by its own elastic force. The geometric shape of the string and its internal stretching configuration evolve in a coupled way, and they combined govern the dynamics of the system. In this paper, we show that certain geometric quantities of the moving string satisfy extremum principles and decay estimates. As a result, we can prove that the 2-D Peskin problem admits a unique global solution when the initial data satisfies a medium-size geometric condition on the string shape, while no assumption on the size of stretching is needed.
We show that the De Gregorio model on the real line admits infinitely many compactly supported, self-similar solutions that are distinct under rescaling and will blow up in finite time. These self-similar solutions fall into two classes: the basic class and the general class. The basic class consists of countably infinite solutions that are eigenfunctions of a self-adjoint compact operator. In particular, the leading eigenfunction coincides with the finite-time singularity solution of the De Gregorio model recently obtained by numerical approaches. The general class consists of more complicated solutions that can be obtained by solving nonlinear eigenvalue problems associated with the same compact operator.
We introduce the tangential Peskin problem in 2-D, which is a scalar drift-diffusion equation with a nonlocal drift. It is derived with a new Eulerian perspective from a special setting of the 2-D Peskin problem where an infinitely long and straight 1-D elastic string deforms tangentially in the Stokes flow induced by itself in the plane. For initial datum in the energy class satisfying natural weak assumptions, we prove existence of its global solutions. This is considered as a super-critical problem in the existing analysis of the Peskin problem based on Lagrangian formulations. Regularity and long-time behavior of the constructed solution is established. Uniqueness of the solution is proved under additional assumptions.
The open domain knowledge base is very important. It is usually extracted from encyclopedia websites and is widely used in knowledge retrieval systems, question answering systems, or recommendation systems. In practice, the key challenge is to maintain an up-to-date knowledge base. Different from Unwieldy fetching all of the data from the encyclopedia dumps, to enlarge the freshness of the knowledge base as big as possible while avoiding invalid fetching, the current knowledge base updating methods usually determine whether entities need to be updated by building a prediction model. However, these methods can only be defined in some specific fields and the result turns out to be obvious bias, due to the problem of data source and data structure. The users' query intentions are often diverse as to the open domain knowledge, so we construct a topic-aware graph network for knowledge updating based on the user query log. Our methods can be summarized as follow: 1. Extract entities through the user's log and select them as seeds 2. Scrape the attributes of seed entities in the encyclopedia website, and self-supervised construct the entity attribute graph for each entity. 3. Use the entity attribute graph to train the GNN entity update model to determine whether the entity needs to be synchronized. 4.Use the encyclopedia knowledge to match and update the filtered entity with the entity in the knowledge base according to the minimum edit times algorithm.
Text classification plays an important role in many practical applications. In the real world, there are extremely small datasets. Most existing methods adopt pre-trained neural network models to handle this kind of dataset. However, these methods are either difficult to deploy on mobile devices because of their large output size or cannot fully extract the deep semantic information between phrases and clauses. This paper proposes a multimodel-based deep learning framework for short-text multiclass classification with an imbalanced and extremely small data set. Our framework mainly includes five layers: The encoder layer uses DISTILBERT to obtain context-sensitive dynamic word vectors that are difficult to represent in traditional feature engineering methods. Since the transformer part of this layer is distilled, our framework is compressed. Then, we use the next two layers to extract deep semantic information. The output of the encoder layer is sent to a bidirectional LSTM network, and the feature matrix is extracted hierarchically through the LSTM at the word and sentence level to obtain the fine-grained semantic representation. After that, the max-pooling layer converts the feature matrix into a lower-dimensional matrix, preserving only the obvious features. Finally, the feature matrix is taken as the input of a fully connected softmax layer, which contains a function that can convert the predicted linear vector into the output value as the probability of the text in each classification. Extensive experiments on two public benchmarks demonstrate the effectiveness of our proposed approach on an extremely small data set. It retains the state-of-the-art baseline performance in terms of precision, recall, accuracy, and F1 score, and through the model size, training time, and convergence epoch, we can conclude that our method can be deployed faster and lighter on mobile devices.
In this paper we study a tumor growth model with nutrients. The model presents dynamic patch solutions due to the contact inhibition among the tumor cells. We show that when the nutrients do not diffuse and the cells do not die, the tumor density exhibits regularizing dynamics. In particular, we provide contraction estimates, exponential rate of asymptotic convergence, and boundary regularity of the tumor patch. These results are in sharp contrast to the models either with nutrient diffusion or with death rate in tumor cells.
In this work, we use the JKO scheme to approximate a general class of diffusion problems generated by Darcy's law. Although the scheme is now classical, if the energy density is spatially inhomogeneous or irregular, many standard methods fail to apply to establish convergence in the continuum limit. To overcome these difficulties, we analyze the scheme through its dual problem and establish a novel $L^1$-contraction principle for the density variable. Notably, the contraction principle relies only on the existence of an optimal transport map and the convexity structure of the energy. As a result, the principle holds in a very general setting, and opens the door to using optimal-transport-based variational schemes to study a larger class of non-linear inhomogeneous parabolic equations.
We introduce a novel variant of the JKO scheme to approximate Darcy's law with a pressure dependent source term. By introducing a new variable that implicitly controls the source term, our scheme is still able to use the standard Wasserstein-2-metric even though the total mass changes over time. Leveraging the dual formulation of our scheme, we show that the discrete-in-time approximations satisfy many useful properties expected for the continuum solutions, such as a comparison principle and uniform $L^1$-equicontinuity. Many of these properties are new even in the well-understood case where the growth term is absent. Finally, we show that our discrete approximations converge to a solution of the corresponding PDE system, including a tumor growth model with a general nonlinear source term.
We study a tumor growth model in two space dimensions, where proliferation of the tumor cells leads to expansion of the tumor domain and migration of surrounding normal tissues into the exterior vacuum. The model features two moving interfaces separating the tumor, the normal tissue, and the exterior vacuum. We prove local-in-time existence and uniqueness of strong solutions for their evolution starting from a nearly radial initial configuration. It is assumed that the tumor has lower mobility than the normal tissue, which is in line with the well-known Saffman-Taylor condition in viscous fingering.
Motivated by Ball and Majumdar's modification of Landau-de Gennes model for nematic liquid crystals, we study energy-minimizer $Q$ of a tensor-valued variational obstacle problem in a bounded 3-D domain with prescribed boundary data. The energy functional is designed to blow up as $Q$ approaches the obstacle. Under certain assumptions, especially on blow-up profile of the singular bulk potential, we prove higher interior regularity of $Q$, and show that the contact set of $Q$ is either empty, or small with characterization of its Hausdorff dimension. We also prove boundary partial regularity of the energy-minimizer.