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In this paper, we initiate the study of isogeometric analysis (IGA) of a quantum three-body problem that has been well-known to be difficult to solve. In the IGA setting, we represent the wavefunctions by linear combinations of B-spline basis functions and solve the problem as a matrix eigenvalue problem. The eigenvalue gives the eigenstate energy while the eigenvector gives the coefficients of the B-splines that lead to the eigenstate. The major difficulty of isogeometric or other finite-element-method-based analyses lies in the lack of boundary conditions and a large number of degrees of freedom for accuracy. For a typical many-body problem with attractive interaction, there are bound and scattering states where bound states have negative eigenvalues. We focus on bound states and start with the analysis for a two-body problem. We demonstrate through various numerical experiments that IGA provides a promising technique to solve the three-body problems.

Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over more traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground state properties of gapped Hamiltonians in finite spatial dimensions, after learning from data obtained by measuring other Hamiltonians in the same quantum phase of matter. In contrast, under widely accepted complexity theory assumptions, classical algorithms that do not learn from data cannot achieve the same guarantee. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases of matter. Our arguments are based on the concept of a classical shadow, a succinct classical description of a many-body quantum state that can be constructed in feasible quantum experiments and be used to predict many properties of the state. Extensive numerical experiments corroborate our theoretical results in a variety of scenarios, including Rydberg atom systems, 2D random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases.

An abundant amount of data gathered during wind tunnel testing and health monitoring of structures inspires the use of machine learning methods to replicate the wind forces. This paper presents a data-driven Gaussian Process-Nonlinear Finite Impulse Response (GP-NFIR) model of the nonlinear self-excited forces acting on structures. Constructed in a nondimensional form, the model takes the effective wind angle of attack as lagged exogenous input and outputs a probability distribution of the forces. The nonlinear input/output function is modeled by a GP regression. Consequently, the model is nonparametric, thereby circumventing to set up the function's structure a priori. The training input is designed as random harmonic motion consisting of vertical and rotational displacements. Once trained, the model can predict the aerodynamic forces for both prescribed input motion and aeroelastic analysis. The concept is first verified for a flat plate's analytical solution by predicting the self-excited forces and flutter velocity. Finally, the framework is applied to a streamlined and bluff bridge deck based on Computational Fluid Dynamics (CFD) data. The model's ability to predict nonlinear aerodynamic forces, flutter velocity, and post-flutter behavior are highlighted. Applications of the framework are foreseen in the structural analysis during the design and monitoring of slender line-like structures.

Determining capacities of quantum channels is a fundamental question in quantum information theory. Despite having rigorous coding theorems quantifying the flow of information across quantum channels, their capacities are poorly understood due to super-additivity effects. Studying these phenomena is important for deepening our understanding of quantum information, yet simple and clean examples of super-additive channels are scarce. Here we study a simple family of qutrit channels called the platypus channel, and show that it exhibits super-additivity of coherent information when used jointly with a variety of qubit channels. A higher-dimensional variant of the platypus channel displays super-additivity of quantum capacity together with an erasure channel. Subject to the "spin-alignment conjecture" introduced in a companion paper, our results on super-additivity of quantum capacity extend to lower-dimensional channels as well as larger parameter ranges. In particular, super-additivity occurs between two weakly additive channels each with large capacity on their own, in stark contrast to previous results. Remarkably, a single, novel transmission strategy achieves super-additivity in all examples. Our results show that super-additivity is much more prevalent than previously thought. It can occur across a wide variety of channels, even when both participating channels have large quantum capacity.

This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.

Deep learning techniques have received much attention in the area of image denoising. However, there are substantial differences in the various types of deep learning methods dealing with image denoising. Specifically, discriminative learning based on deep learning can ably address the issue of Gaussian noise. Optimization models based on deep learning are effective in estimating the real noise. However, there has thus far been little related research to summarize the different deep learning techniques for image denoising. In this paper, we offer a comparative study of deep techniques in image denoising. We first classify the deep convolutional neural networks (CNNs) for additive white noisy images; the deep CNNs for real noisy images; the deep CNNs for blind denoising and the deep CNNs for hybrid noisy images, which represents the combination of noisy, blurred and low-resolution images. Then, we analyze the motivations and principles of the different types of deep learning methods. Next, we compare the state-of-the-art methods on public denoising datasets in terms of quantitative and qualitative analysis. Finally, we point out some potential challenges and directions of future research.

Quantum hardware and quantum-inspired algorithms are becoming increasingly popular for combinatorial optimization. However, these algorithms may require careful hyperparameter tuning for each problem instance. We use a reinforcement learning agent in conjunction with a quantum-inspired algorithm to solve the Ising energy minimization problem, which is equivalent to the Maximum Cut problem. The agent controls the algorithm by tuning one of its parameters with the goal of improving recently seen solutions. We propose a new Rescaled Ranked Reward (R3) method that enables stable single-player version of self-play training that helps the agent to escape local optima. The training on any problem instance can be accelerated by applying transfer learning from an agent trained on randomly generated problems. Our approach allows sampling high-quality solutions to the Ising problem with high probability and outperforms both baseline heuristics and a black-box hyperparameter optimization approach.

Graph convolution is the core of most Graph Neural Networks (GNNs) and usually approximated by message passing between direct (one-hop) neighbors. In this work, we remove the restriction of using only the direct neighbors by introducing a powerful, yet spatially localized graph convolution: Graph diffusion convolution (GDC). GDC leverages generalized graph diffusion, examples of which are the heat kernel and personalized PageRank. It alleviates the problem of noisy and often arbitrarily defined edges in real graphs. We show that GDC is closely related to spectral-based models and thus combines the strengths of both spatial (message passing) and spectral methods. We demonstrate that replacing message passing with graph diffusion convolution consistently leads to significant performance improvements across a wide range of models on both supervised and unsupervised tasks and a variety of datasets. Furthermore, GDC is not limited to GNNs but can trivially be combined with any graph-based model or algorithm (e.g. spectral clustering) without requiring any changes to the latter or affecting its computational complexity. Our implementation is available online.

This paper tackles a new problem setting: reinforcement learning with pixel-wise rewards (pixelRL) for image processing. After the introduction of the deep Q-network, deep RL has been achieving great success. However, the applications of deep RL for image processing are still limited. Therefore, we extend deep RL to pixelRL for various image processing applications. In pixelRL, each pixel has an agent, and the agent changes the pixel value by taking an action. We also propose an effective learning method for pixelRL that significantly improves the performance by considering not only the future states of the own pixel but also those of the neighbor pixels. The proposed method can be applied to some image processing tasks that require pixel-wise manipulations, where deep RL has never been applied. We apply the proposed method to three image processing tasks: image denoising, image restoration, and local color enhancement. Our experimental results demonstrate that the proposed method achieves comparable or better performance, compared with the state-of-the-art methods based on supervised learning.

Degradation of image quality due to the presence of haze is a very common phenomenon. Existing DehazeNet [3], MSCNN [11] tackled the drawbacks of hand crafted haze relevant features. However, these methods have the problem of color distortion in gloomy (poor illumination) environment. In this paper, a cardinal (red, green and blue) color fusion network for single image haze removal is proposed. In first stage, network fusses color information present in hazy images and generates multi-channel depth maps. The second stage estimates the scene transmission map from generated dark channels using multi channel multi scale convolutional neural network (McMs-CNN) to recover the original scene. To train the proposed network, we have used two standard datasets namely: ImageNet [5] and D-HAZY [1]. Performance evaluation of the proposed approach has been carried out using structural similarity index (SSIM), mean square error (MSE) and peak signal to noise ratio (PSNR). Performance analysis shows that the proposed approach outperforms the existing state-of-the-art methods for single image dehazing.