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Steepest descent algorithms, which are commonly used in deep learning, use the gradient as the descent direction, either as-is or after a direction shift using preconditioning. In many scenarios calculating the gradient is numerically hard…

We present a second-order strictly length-preserving and unconditionally energy-stable rotational discrete gradient (Rdg) scheme for the numerical approximation of the Oseen-Frank gradient flows with anisotropic elastic energy functional.…

Machine-learned regression models represent a promising tool to implement accurate and computationally affordable energy-density functionals to solve quantum many-body problems via density functional theory. However, while they can easily…

Accelerated gradient descent iterations are widely used in optimization. It is known that, in the continuous-time limit, these iterations converge to a second-order differential equation which we refer to as the accelerated gradient flow.…

Accurate and efficient fluid flow models are essential for applications relating to many physical phenomena including geophysical, aerodynamic, and biological systems. While these flows may exhibit rich and multiscale dynamics, in many…

The path signature, having enjoyed recent success in the machine learning community, is a theoretically-driven method for engineering features from irregular paths. On the other hand, graph neural networks (GNN), neural architectures for…

Dual gradient descent combined with early stopping represents an efficient alternative to the Tikhonov variational approach when the regularizer is strongly convex. However, for many relevant applications, it is crucial to deal with…

Deep learning-based algorithms have greatly improved the performance of remote sensing image (RSI) super-resolution (SR). However, increasing network depth and parameters cause a huge burden of computing and storage. Directly reducing the…

We develop a model reduction technique for non-smooth dynamical systems using spectral submanifolds. Specifically, we construct low-dimensional, sparse, nonlinear and non-smooth models on unions of slow and attracting spectral submanifolds…

With more energy networks being interconnected to form integrated energy systems (IESs), the optimal energy flow (OEF) problem has drawn increasing attention. Extant studies on OEF models mostly utilize the finite difference method (FDM) to…

This paper studies a class of linear unconditionally energy stable schemes for the gradient flows. Such schemes are built on the SAV technique and the general linear time discretization (GLTD) as well as the linearization based on the…

Reinforcement learning (RL) has shown great potential in training agile and adaptable controllers for legged robots, enabling them to learn complex locomotion behaviors directly from experience. However, policies trained in simulation often…

We propose an end-to-end framework based on a Graph Neural Network (GNN) to balance the power flows in energy grids. The balancing is framed as a supervised vertex regression task, where the GNN is trained to predict the current and power…

This study aims to reconstruct the complete flow field from spatially restricted domain data by utilizing an Enhanced Super-Resolution Generative Adversarial Network (ESRGAN) model. The difficulty in flow field reconstruction lies in…

Traditional ground-penetrating radar (GPR) data inversion leverages iterative algorithms which suffer from high computation costs and low accuracy when applied to complex subsurface scenarios. Existing deep learning-based methods focus on…

While significant theoretical progress has been achieved, unveiling the generalization mystery of overparameterized neural networks still remains largely elusive. In this paper, we study the generalization behavior of shallow neural…

The Similarity Renormalization Group (SRG) is a continuous series of unitary transformations that can be implemented as a flow equation. When the relative kinetic energy ($\Trel$) is used in the SRG generator, nuclear structure calculations…

The operator-theoretic renormalization group (RG) methods are powerful analytic tools to explore spectral properties of field-theoretical models such as quantum electrodynamics (QED) with non-relativistic matter. In this paper these methods…

The increasing demand for electricity, coupled with the rise in greenhouse gas emissions, necessitates the integration of Renewable Energy Sources (RESs) into power grids. However, the fluctuating nature of RESs introduces new challenges in…

In recent years, the scalar auxiliary variable (SAV) approach has become very popular and hot in the design of linear, high-order and unconditional energy stable schemes of gradient flow models. However, the nature of SAV-based numerical…