Re: [WORK] Differential Geometry By Mittal And Agarwal Pdf Download
Cherly Fleitas
Research Areas of our faculty members include: Theoretical and numerical analysis of ODEs & PDEs, Mathematical modelling, Fluid dynamics, Non-equilibrium thermodynamics, Continuum mechanics, Applied Probability Theory, Queueing theory, Stochastic processes, Statistical inference, Fuzzy theory, Algebra, Algebraic geometry, Number theory, Cosmology, Graph theory, Computational design, Neural networks, Fractional calculus, Geometric function theory.
Variational methods also work quite well for image denoising. In variational methods one considers the minimizer of a suitable functional, which usually satisfies a partial differential equation. Then we use the minimizer as the denoised image. This is one way to study the inverse problem mentioned at the beginning of this paper. The most popular functional used in image processing are the Dirichlet functionaland the TV functionalAccording to these functionals there are models and TV models for image denoising. Particularly the TV model proposed by Rudin, Osher, and Fatemi [1] is very useful in image processing.
Modeling sequential patterns from data is at the core of various time series forecasting tasks. Deep learning models have greatly outperformed many traditional models, but these black-box models generally lack explainability in prediction and decision making. To reveal the underlying trend with understandable mathematical expressions, scientists and economists tend to use partial differential equations (PDEs) to explain the highly nonlinear dynamics of sequential patterns. However, it usually requires domain expert knowledge and a series of simplified assumptions, which is not always practical and can deviate from the ever-changing world. Is it possible to learn the differential relations from data dynamically to explain the time-evolving dynamics? In this work, we propose an learning framework that can automatically obtain interpretable PDE models from sequential data. Particularly, this framework is comprised of learnable differential blocks, named P-blocks, which is proved to be able to approximate any time-evolving complex continuous functions in theory. Moreover, to capture the dynamics shift, this framework introduces a meta-learning controller to dynamically optimize the hyper-parameters of a hybrid PDE model. Extensive experiments on times series forecasting of financial, engineering, and health data show that our model can provide valuable interpretability and achieve comparable performance to state-of-the-art models. From empirical studies, we find that learning a few differential operators may capture the major trend of sequential dynamics without massive computational complexity.
Opinion formation and propagation are crucial phenomena in social networks and have been extensively studied across several disciplines. Traditionally, theoretical models of opinion dynamics have been proposed to describe the interactions between individuals (i.e., social interaction) and their impact on the evolution of collective opinions. Although these models can incorporate sociological and psychological knowledge on the mechanisms of social interaction, they demand extensive calibration with real data to make reliable predictions, requiring much time and effort. Recently, the widespread use of social media platforms provides new paradigms to learn deep learning models from a large volume of social media data. However, these methods ignore any scientific knowledge about the mechanism of social interaction. In this work, we present the first hybrid method called Sociologically-Informed Neural Network (SINN), which integrates theoretical models and social media data by transporting the concepts of physics-informed neural networks (PINNs) from natural science (i.e., physics) into social science (i.e., sociology and social psychology). In particular, we recast theoretical models as ordinary differential equations (ODEs). Then we train a neural network that simultaneously approximates the data and conforms to the ODEs that represent the social scientific knowledge. In addition, we extend PINNs by integrating matrix factorization and a language model to incorporate rich side information (e.g., user profiles) and structural knowledge (e.g., cluster structure of the social interaction network). Moreover, we develop an end-to-end training procedure for SINN, which involves Gumbel-Softmax approximation to include stochastic mechanisms of social interaction. Extensive experiments on real-world and synthetic datasets show SINN outperforms six baseline methods in predicting opinion dynamics.
Topic mining extracts patterns and insights from text data (e.g., documents, emails and product reviews), which can be used in various applications such as intent detection. However, topic mining can result in severe privacy threats to the users who have contributed to the text corpus since they can be re-identified from the text data with certain background knowledge. To our best knowledge, we propose the first differentially private topic mining technique (namely TopicDP) which injects well-calibrated Gaussian noise into the matrix output of any topic mining algorithm to ensure differential privacy and good utility. Specifically, we smoothen the sensitivity for the Gaussian mechanism via sensitivity sampling, which addresses the major challenges resulted from the high sensitivity in topic mining for differential privacy. Furthermore, we theoretically prove the differential privacy guarantee under the Rényi differential privacy mechanism and the utility error bounds of TopicDP. Finally, we conduct extensive experiments on two real-word text datasets (Enron email and Amazon Reviews), and the experimental results demonstrate that TopicDP is a model-agnostic framework that can generate better privacy preserving performance for topic mining as compared against other differential privacy mechanisms.
Graph neural networks (GNN) are powerful tools in many web research problems. However, existing GNNs are not fully suitable for many real-world web applications. For example, over-smoothing may affect personalized recommendations and the lack of an explanation for the GNN prediction hind the understanding of many business scenarios. To address these problems, in this paper, we propose a new second-order continuous GNN which naturally avoids over-smoothing and enjoys better interpretability. There is some research interest in continuous graph neural networks inspired by the recent success of neural ordinary differential equations (ODEs). However, there are some remaining problems w.r.t. the prevailing first-order continuous GNN frameworks. Firstly, augmenting node features is an essential, however heuristic step for the numerical stability of current frameworks; secondly, first-order methods characterize a diffusion process, in which the over-smoothing effect w.r.t. node representations are intrinsic; and thirdly, there are some difficulties to integrate the topology of graphs into the ODEs. Therefore, we propose a framework employing second-order graph neural networks, which usually learn a less stiff transformation than the first-order counterpart. Our method can also be viewed as a coupled first-order model, which is easy to implement. We propose a semi-model-agnostic method based on our model to enhance the prediction explanation using high-order information. We construct an analog between continuous GNNs and some famous partial differential equations and discuss some properties of the first and second-order models. Extensive experiments demonstrate the effectiveness of our proposed method, and the results outperform related baselines.
To this effect, we focus on discovery of the "game behaviours" as micro-patterns formed by continuous sequence of games and the persistent "play styles" of the players' as a sequence of such sequences on an online skill gaming platform for Rummy. The complex sequences of intricate sequences is analysed through a novel collaborative two stage deep neural network, CognitionNet. The first stage focuses on mining game behaviours as cluster representations in a latent space while the second aggregates over these micro patterns (e.g., transitions across patterns) to discover play styles via a supervised classification objective around player engagement. The dual objective allows CognitionNet to reveal several player psychology inspired decision making and tactics. To our knowledge, this is the first and one-of-its-kind research to fully automate the discovery of: (i) player psychology and game tactics from telemetry data; and (ii) relevant diagnostic explanations to players' engagement predictions. The collaborative training of the two networks with differential input dimensions is enabled using a novel formulation of "bridge loss". The network plays pivotal role in obtaining homogeneous and consistent play style definitions and significantly outperforms the SOTA baselines wherever applicable.
For those seeking healthcare advice online, AI based dialogue agents capable of interacting with patients to perform automatic disease diagnosis are a viable option. This application necessitates efficient inquiry of relevant disease symptoms in order to make accurate diagnosis recommendations. This can be formulated as a problem of sequential feature (symptom) selection and classification for which reinforcement learning (RL) approaches have been proposed as a natural solution. They perform well when the feature space is small, that is, the number of symptoms and diagnosable disease categories is limited, but they frequently fail in assignments with a large number of features. To address this challenge, we propose a Multi-Model-Fused Actor-Critic (MMF-AC) RL framework that consists of a generative actor network and a diagnostic critic network. The actor incorporates a Variational AutoEncoder (VAE) to model the uncertainty induced by partial observations of features, thereby facilitating in making appropriate inquiries. In the critic network, a supervised diagnosis model for disease predictions is involved to precisely estimate the state-value function. Furthermore, inspired by the medical concept of differential diagnosis, we combine the generative and diagnosis models to create a novel reward shaping mechanism to address the sparse reward problem in large search spaces. We conduct extensive experiments on both synthetic and real-world datasets for empirical evaluations. The results demonstrate that our approach outperforms state-of-the-art methods in terms of diagnostic accuracy and interaction efficiency while also being more effectively scalable to large search spaces. Besides, our method is adaptable to both categorical and continuous features, making it ideal for online applications.
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