Non-Hermitian (NH) systems may have quite different physics properties from that of Hermitian counterparts. For example, in the NH systems with nonreciprocal hopping, there exists (single-body version of) skin effect-The eigenstates are exponentially
localized at the boundaries. An interesting problem is about the generalization of NH skin effect to a many-body NH system. In this paper, we studied many-body physics in the quantum systems with nonreciprocal hoppings and obtained analytical results. In these many-body NH systems, the single-body NH skin effect upgrades to quantum (Maxwell s) pressure-demon effect, which leads to band-width renormalization and a uniform pressure gradient to the system. In particular, according to the quantum pressure-demon effect, in many-body Bosonic/Fermionic Hotano-Nelson model, there exist new physical phenomena compared with their Hermitian counterparts: Liouvillian Bose-Einstein condensation and Liouvillian Fermi-surface in real space, respectively. This discovery will open a door to learn many-body physics for NH quantum systems.
We develop a spacetime neural network method with second order optimization for solving quantum dynamics from the high dimensional Schr{o}dinger equation. In contrast to the standard iterative first order optimization and the time-dependent variation
al principle, our approach utilizes the implicit mid-point method and generates the solution for all spatial and temporal values simultaneously after optimization. We demonstrate the method in the Schr{o}dinger equation with a self-normalized autoregressive spacetime neural network construction. Future explorations for solving different high dimensional differential equations are discussed.
This paper considers two-player zero-sum finite-horizon Markov games with simultaneous moves. The study focuses on the challenging settings where the value function or the model is parameterized by general function classes. Provably efficient algorit
hms for both decoupled and {coordinated} settings are developed. In the {decoupled} setting where the agent controls a single player and plays against an arbitrary opponent, we propose a new model-free algorithm. The sample complexity is governed by the Minimax Eluder dimension -- a new dimension of the function class in Markov games. As a special case, this method improves the state-of-the-art algorithm by a $sqrt{d}$ factor in the regret when the reward function and transition kernel are parameterized with $d$-dimensional linear features. In the {coordinated} setting where both players are controlled by the agent, we propose a model-based algorithm and a model-free algorithm. In the model-based algorithm, we prove that sample complexity can be bounded by a generalization of Witness rank to Markov games. The model-free algorithm enjoys a $sqrt{K}$-regret upper bound where $K$ is the number of episodes. Our algorithms are based on new techniques of alternate optimism.
At a macroscopic level, concepts such as top spin, back spin and rolling are commonly used to describe the collision of balls and surfaces. Each term refers to an aspect of the coupling of rotational motion during the collision of a spherical particl
e with a planar surface. In this paper we explore the mechanisms of energy transfer involving the collision of a rotating sphere and a surface using a model of frictional interactions developed for granular material. We present explicit analytical treatments for the scattering and derive expressions for two important limiting classes: energy conserving collisions and collisions subject to rapid transverse dissipation.
General Value Function (GVF) is a powerful tool to represent both the {em predictive} and {em retrospective} knowledge in reinforcement learning (RL). In practice, often multiple interrelated GVFs need to be evaluated jointly with pre-collected off-p
olicy samples. In the literature, the gradient temporal difference (GTD) learning method has been adopted to evaluate GVFs in the off-policy setting, but such an approach may suffer from a large estimation error even if the function approximation class is sufficiently expressive. Moreover, none of the previous work have formally established the convergence guarantee to the ground truth GVFs under the function approximation settings. In this paper, we address both issues through the lens of a class of GVFs with causal filtering, which cover a wide range of RL applications such as reward variance, value gradient, cost in anomaly detection, stationary distribution gradient, etc. We propose a new algorithm called GenTD for off-policy GVFs evaluation and show that GenTD learns multiple interrelated multi-dimensional GVFs as efficiently as a single canonical scalar value function. We further show that unlike GTD, the learned GVFs by GenTD are guaranteed to converge to the ground truth GVFs as long as the function approximation power is sufficiently large. To our best knowledge, GenTD is the first off-policy GVF evaluation algorithm that has global optimality guarantee.
Multiple views of data, both naturally acquired (e.g., image and audio) and artificially produced (e.g., via adding different noise to data samples), have proven useful in enhancing representation learning. Natural views are often handled by multivie
w analysis tools, e.g., (deep) canonical correlation analysis [(D)CCA], while the artificial ones are frequently used in self-supervised learning (SSL) paradigms, e.g., SimCLR and Barlow Twins. Both types of approaches often involve learning neural feature extractors such that the embeddings of data exhibit high cross-view correlations. Although intuitive, the effectiveness of correlation-based neural embedding is only empirically validated. This work puts forth a theory-backed framework for unsupervised multiview learning. Our development starts with proposing a multiview model, where each view is a nonlinear mixture of shared and private components. Consequently, the learning problem boils down to shared/private component identification and disentanglement. Under this model, latent correlation maximization is shown to guarantee the extraction of the shared components across views (up to certain ambiguities). In addition, the private information in each view can be provably disentangled from the shared using proper regularization design. The method is tested on a series of tasks, e.g., downstream clustering, which all show promising performance. Our development also provides a unifying perspective for understanding various DCCA and SSL schemes.
Image-text matching plays a central role in bridging the semantic gap between vision and language. The key point to achieve precise visual-semantic alignment lies in capturing the fine-grained cross-modal correspondence between image and text. Most p
revious methods rely on single-step reasoning to discover the visual-semantic interactions, which lacks the ability of exploiting the multi-level information to locate the hierarchical fine-grained relevance. Different from them, in this work, we propose a step-wise hierarchical alignment network (SHAN) that decomposes image-text matching into multi-step cross-modal reasoning process. Specifically, we first achieve local-to-local alignment at fragment level, following by performing global-to-local and global-to-global alignment at context level sequentially. This progressive alignment strategy supplies our model with more complementary and sufficient semantic clues to understand the hierarchical correlations between image and text. The experimental results on two benchmark datasets demonstrate the superiority of our proposed method.
In the current control design of safety-critical autonomous systems, formal verification techniques are typically applied after the controller is designed to evaluate whether the required properties (e.g., safety) are satisfied. However, due to the i
ncreasing system complexity and the fundamental hardness of designing a controller with formal guarantees, such an open-loop process of design-then-verify often results in many iterations and fails to provide the necessary guarantees. In this paper, we propose a correct-by-construction control learning framework that integrates the verification into the control design process in a closed-loop manner, i.e., design-while-verify. Specifically, we leverage the verification results (computed reachable set of the system state) to construct feedback metrics for control learning, which measure how likely the current design of control parameters can meet the required reach-avoid property for safety and goal-reaching. We formulate an optimization problem based on such metrics for tuning the controller parameters, and develop an approximated gradient descent algorithm with a difference method to solve the optimization problem and learn the controller. The learned controller is formally guaranteed to meet the required reach-avoid property. By treating verifiability as a first-class objective and effectively leveraging the verification results during the control learning process, our approach can significantly improve the chance of finding a control design with formal property guarantees. This is demonstrated via a set of experiments on both linear and non-linear systems that use model-based or neural network based controllers.
In this paper, we propose a local-global multiscale method for highly heterogeneous stochastic groundwater flow problems under the framework of reduced basis method and the generalized multiscale finite element method (GMsFEM). Due to incomplete char
acterization of the medium properties of the groundwater flow problems, random variables are used to parameterize the uncertainty. As a result, solving the problem repeatedly is required to obtain statistical quantities. Besides, the medium properties are usually highly heterogeneous, which will result in a large linear system that needs to be solved. Therefore, it is intrinsically inevitable to seek a computational-efficient model reduction method to overcome the difficulty. We will explore the combination of the reduced basis method and the GMsFEM. In particular, we will use residual-driven basis functions, which are key ingredients in GMsFEM. This local-global multiscale method is more efficient than applying the GMsFEM or reduced basis method individually. We first construct parameter-independent multiscale basis functions that include both local and global information of the permeability fields, and then use these basis functions to construct several global snapshots and global basis functions for fast online computation with different parameter inputs. We provide rigorous analysis of the proposed method and extensive numerical examples to demonstrate the accuracy and efficiency of the local-global multiscale method.
Let $X:={X(t)}_{tge0}$ be a generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019): $$ big{X(t)big}_{tge0}overset{d}{=}left{ int_{mathbb R} left((t-u)_+^{alpha}-(-u)_+^{alpha} right) |u|^{-gamma} B(du) right}_{tge0},
$$ with parameters $gamma in (0, 1/2)$ and $alphain left(-frac12+ gamma , , frac12+ gamma right)$. Continuing the studies of sample path properties of GFBM $X$ in Ichiba, Pang and Taqqu (2021) and Wang and Xiao (2021), we establish integral criteria for the lower functions of $X$ at $t=0$ and at infinity by modifying the arguments of Talagrand (1996). As a consequence of the integral criteria, we derive the Chung-type laws of the iterated logarithm of $X$ at the $t=0$ and at infinity, respectively. This solves a problem in Wang and Xiao (2021).
