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One of the central questions in the theory of deep learning is to understand how neural networks learn hierarchical features. The ability of deep networks to extract salient features is crucial to both their outstanding generalization ability and the modern deep learning paradigm of pretraining and finetuneing. However, this feature learning process remains poorly understood from a theoretical perspective, with existing analyses largely restricted to two-layer networks. In this work we show that three-layer neural networks have provably richer feature learning capabilities than two-layer networks. We analyze the features learned by a three-layer network trained with layer-wise gradient descent, and present a general purpose theorem which upper bounds the sample complexity and width needed to achieve low test error when the target has specific hierarchical structure. We instantiate our framework in specific statistical learning settings -- single-index models and functions of quadratic features -- and show that in the latter setting three-layer networks obtain a sample complexity improvement over all existing guarantees for two-layer networks. Crucially, this sample complexity improvement relies on the ability of three-layer networks to efficiently learn nonlinear features. We then establish a concrete optimization-based depth separation by constructing a function which is efficiently learnable via gradient descent on a three-layer network, yet cannot be learned efficiently by a two-layer network. Our work makes progress towards understanding the provable benefit of three-layer neural networks over two-layer networks in the feature learning regime.

In this paper, we propose a new deep learning method, named finite volume method (DFVM) to solve high-dimension partial differential equations (PDEs). The key idea of DFVM is that we construct a new loss function under the framework of the finite volume method. The weak formulation makes DFVM more feasible to solve general high dimensional PDEs defined on arbitrarily shaped domains. Numerical solutions obtained by DFVM also enjoy physical conservation property in the control volume of each sampling point, which is not available in other existing deep learning methods. Numerical results illustrate that DFVM not only reduces the computation cost but also obtains more accurate approximate solutions. Specifically, for high-dimensional linear and nonlinear elliptic PDEs, DFVM provides better approximations than DGM and WAN, by one order of magnitude. The relative error obtained by DFVM is slightly smaller than that obtained by PINN, but the computation cost of DFVM is an order of magnitude less than that of the PINN. For the time-dependent Black-Scholes equation, DFVM gives better approximations than PINN, by one order of magnitude.

Group testing enables to identify infected individuals in a population using a smaller number of tests than individual testing. To achieve this, group testing algorithms commonly assume knowledge of the number of infected individuals; nonadaptive and several adaptive algorithms fall in this category. Some adaptive algorithms, like binary splitting, operate without this assumption, but require a number of stages that may scale linearly with the size of the population. In this paper we contribute a new algorithm that enables a balance between the number of tests and the number of stages used, and which we term diagonal group testing. Diagonal group testing, like binary splitting, does not require knowledge of the number of infected individuals, yet unlike binary splitting, is order-optimal w.r.t. the expected number of tests it requires and is guaranteed to succeed in a small number of stages that scales at most logarithmically with the size of the population. Numerical evaluations, for diagonal group testing and a hybrid approach we propose, support our theoretical findings.

An adaptive modified weak Galerkin method (AmWG) for an elliptic problem is studied in this paper, in addition to its convergence and optimality. The modified weak Galerkin bilinear form is simplified without the need of the skeletal variable, and the approximation space is chosen as the discontinuous polynomial space as in the discontinuous Galerkin method. Upon a reliable residual-based a posteriori error estimator, an adaptive algorithm is proposed together with its convergence and quasi-optimality proved for the lowest order case. The primary tool is to bridge the connection between the modified weak Galerkin method and the Crouzeix-Raviart nonconforming finite element. Unlike the traditional convergence analysis for methods with a discontinuous polynomial approximation space, the convergence of AmWG is penalty parameter free. Numerical results are presented to support the theoretical results.

Neural network (NN) designed for challenging machine learning tasks is in general a highly nonlinear mapping that contains massive variational parameters. High complexity of NN, if unbounded or unconstrained, might unpredictably cause severe issues including over-fitting, loss of generalization power, and unbearable cost of hardware. In this work, we propose a general compression scheme that significantly reduces the variational parameters of NN by encoding them to multi-layer tensor networks (TN's) that contain exponentially-fewer free parameters. Superior compression performance of our scheme is demonstrated on several widely-recognized NN's (FC-2, LeNet-5, and VGG-16) and datasets (MNIST and CIFAR-10), surpassing the state-of-the-art method based on shallow tensor networks. For instance, about 10 million parameters in the three convolutional layers of VGG-16 are compressed in TN's with just $632$ parameters, while the testing accuracy on CIFAR-10 is surprisingly improved from $81.14\%$ by the original NN to $84.36\%$ after compression. Our work suggests TN as an exceptionally efficient mathematical structure for representing the variational parameters of NN's, which superiorly exploits the compressibility than the simple multi-way arrays.

The detection of Maximal Extractable Value (MEV) in blockchain is crucial for enhancing blockchain security, as it enables the evaluation of potential consensus layer risks, the effectiveness of anti-centralization solutions, and the assessment of user exploitation. However, existing MEV detection methods face limitations due to their low recall rate, reliance on pre-registered Application Binary Interfaces (ABIs) and the need for continuous monitoring of new DeFi services. In this paper, we propose ArbiNet, a novel GNN-based detection model that offers a low-overhead and accurate solution for MEV detection without requiring knowledge of smart contract code or ABIs. We collected an extensive MEV dataset, surpassing currently available public datasets, to train ArbiNet. Our implemented model and open dataset enhance the understanding of the MEV landscape, serving as a foundation for MEV quantification and improved blockchain security.

We introduce a priori Sobolev-space error estimates for the solution of nonlinear, and possibly parametric, PDEs using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.

The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin. Traditional parameterised differential equations are a special case. Many popular neural network architectures, such as residual networks and recurrent networks, are discretisations. NDEs are suitable for tackling generative problems, dynamical systems, and time series (particularly in physics, finance, ...) and are thus of interest to both modern machine learning and traditional mathematical modelling. NDEs offer high-capacity function approximation, strong priors on model space, the ability to handle irregular data, memory efficiency, and a wealth of available theory on both sides. This doctoral thesis provides an in-depth survey of the field. Topics include: neural ordinary differential equations (e.g. for hybrid neural/mechanistic modelling of physical systems); neural controlled differential equations (e.g. for learning functions of irregular time series); and neural stochastic differential equations (e.g. to produce generative models capable of representing complex stochastic dynamics, or sampling from complex high-dimensional distributions). Further topics include: numerical methods for NDEs (e.g. reversible differential equations solvers, backpropagation through differential equations, Brownian reconstruction); symbolic regression for dynamical systems (e.g. via regularised evolution); and deep implicit models (e.g. deep equilibrium models, differentiable optimisation). We anticipate this thesis will be of interest to anyone interested in the marriage of deep learning with dynamical systems, and hope it will provide a useful reference for the current state of the art.

Learning node embeddings that capture a node's position within the broader graph structure is crucial for many prediction tasks on graphs. However, existing Graph Neural Network (GNN) architectures have limited power in capturing the position/location of a given node with respect to all other nodes of the graph. Here we propose Position-aware Graph Neural Networks (P-GNNs), a new class of GNNs for computing position-aware node embeddings. P-GNN first samples sets of anchor nodes, computes the distance of a given target node to each anchor-set,and then learns a non-linear distance-weighted aggregation scheme over the anchor-sets. This way P-GNNs can capture positions/locations of nodes with respect to the anchor nodes. P-GNNs have several advantages: they are inductive, scalable,and can incorporate node feature information. We apply P-GNNs to multiple prediction tasks including link prediction and community detection. We show that P-GNNs consistently outperform state of the art GNNs, with up to 66% improvement in terms of the ROC AUC score.

Deep learning has emerged as a powerful machine learning technique that learns multiple layers of representations or features of the data and produces state-of-the-art prediction results. Along with the success of deep learning in many other application domains, deep learning is also popularly used in sentiment analysis in recent years. This paper first gives an overview of deep learning and then provides a comprehensive survey of its current applications in sentiment analysis.