arXiv subjects

Zhenyuan Zhang

Publications and source records attributed to Zhenyuan Zhang.

49 records · Page 3Linked to original sources

arXiv2024

Modeling Shortest Paths in Polymeric Networks using Spatial Branching Processes

Recent studies have established a connection between the macroscopic mechanical response of polymeric materials and the statistics of the shortest path (SP) length between distant nodes in the polymer network. Since these statistics can be costly to compute and difficult to study theoretically, we introduce a branching random walk (BRW) model to describe the SP statistics from the coarse-grained molecular dynamics (CGMD) simulations of polymer networks. We postulate that the first passage time (FPT) of the BRW to a given termination site can be used to approximate the statistics of the SP between distant nodes in the polymer network. We develop a theoretical framework for studying the FPT of spatial branching processes and obtain an analytical expression for estimating the FPT distribution as a function of the cross-link density. We demonstrate by extensive numerical calculations that the distribution of the FPT of the BRW model agrees well with the SP distribution from the CGMD simulations. The theoretical estimate and the corresponding numerical implementations of BRW provide an efficient way of approximating the SP distribution in a polymer network. Our results have the physical meaning that by accounting for the realistic topology of polymer networks, extensive bond-breaking is expected to occur at a much smaller stretch than that expected from idealized models assuming periodic network structures. Our work presents the first analysis of polymer networks as a BRW and sets the framework for developing a generalizable spatial branching model for studying the macroscopic evolution of polymeric systems.

math-ph↗

arXiv2023

Weierstrass Bridges

We introduce a new class of stochastic processes called fractional Wiener-Weierstrass bridges. They arise by applying the convolution from the construction of the classical, fractal Weierstrass functions to an underlying fractional Brownian bridge. By analyzing the $p$-th variation of the fractional Wiener-Weierstrass bridge along the sequence of $b$-adic partitions, we identify two regimes in which the processes exhibit distinct sample path properties. We also analyze the critical case between those two regimes for Wiener-Weierstrass bridges that are based on standard Brownian bridge. We furthermore prove that fractional Wiener-Weierstrass bridges are never semimartingales, and we show that their covariance functions are typically fractal functions. Some of our results are extended to Weierstrass bridges based on bridges derived from a general continuous Gaussian martingale.

math.PR↗

arXiv2023

RWKV: Reinventing RNNs for the Transformer Era

Transformers have revolutionized almost all natural language processing (NLP) tasks but suffer from memory and computational complexity that scales quadratically with sequence length. In contrast, recurrent neural networks (RNNs) exhibit linear scaling in memory and computational requirements but struggle to match the same performance as Transformers due to limitations in parallelization and scalability. We propose a novel model architecture, Receptance Weighted Key Value (RWKV), that combines the efficient parallelizable training of transformers with the efficient inference of RNNs. Our approach leverages a linear attention mechanism and allows us to formulate the model as either a Transformer or an RNN, thus parallelizing computations during training and maintains constant computational and memory complexity during inference. We scale our models as large as 14 billion parameters, by far the largest dense RNN ever trained, and find RWKV performs on par with similarly sized Transformers, suggesting future work can leverage this architecture to create more efficient models. This work presents a significant step towards reconciling trade-offs between computational efficiency and model performance in sequence processing tasks.

cs.CL↗

arXiv2021

Sparse-softmax: A Simpler and Faster Alternative Softmax Transformation

The softmax function is widely used in artificial neural networks for the multiclass classification problems, where the softmax transformation enforces the output to be positive and sum to one, and the corresponding loss function allows to use maximum likelihood principle to optimize the model. However, softmax leaves a large margin for loss function to conduct optimizing operation when it comes to high-dimensional classification, which results in low-performance to some extent. In this paper, we provide an empirical study on a simple and concise softmax variant, namely sparse-softmax, to alleviate the problem that occurred in traditional softmax in terms of high-dimensional classification problems. We evaluate our approach in several interdisciplinary tasks, the experimental results show that sparse-softmax is simpler, faster, and produces better results than the baseline models.

cs.LG↗

arXiv2021

A limit theorem for Bernoulli convolutions and the $Φ$-variation of functions in the Takagi class

We consider a probabilistic approach to compute the Wiener--Young $Φ$-variation of fractal functions in the Takagi class. Here, the $Φ$-variation is understood as a generalization of the quadratic variation or, more generally, the $p^{\text{th}}$ variation of a trajectory computed along the sequence of dyadic partitions of the unit interval. The functions $Φ$ we consider form a very wide class of functions that are regularly varying at zero. Moreover, for each such function $Φ$, our results provide in a straightforward manner a large and tractable class of functions that have nontrivial and linear $Φ$-variation. As a corollary, we also construct stochastic processes whose sample paths have nontrivial, deterministic, and linear $Φ$-variation for each function $Φ$ from our class. The proof of our main result relies on a limit theorem for certain sums of Bernoulli random variables that converge to an infinite Bernoulli convolution.

math.PR↗

arXiv2021

Flat band carrier confinement in magic-angle twisted bilayer graphene

Magic angle twisted bilayer graphene has emerged as a powerful platform for studying strongly correlated electron physics, owing to its almost dispersionless low-energy bands and the ability to tune the band filling by electrostatic gating. Techniques to control the twist angle between graphene layers have led to rapid experimental progress but improving sample quality is essential for separating the delicate correlated-electron physics from disorder effects. Owing to the 2D nature of the system and the relatively low carrier density, the samples are highly susceptible to small doping inhomogeneity which can drastically modify the local potential landscape. This potential disorder is distinct from the twist-angle variation which has been studied elsewhere. Here, by using low temperature scanning tunneling spectroscopy and planar tunneling junction measurements, we demonstrate that flat bands in twisted bilayer graphene can amplify small doping inhomogeneity that surprisingly leads to carrier confinement, which in graphene could previously only be realized in the presence of a strong magnetic field.

cond-mat.mes-hall↗

arXiv2021

TAMPC: A Controller for Escaping Traps in Novel Environments

We propose an approach to online model adaptation and control in the challenging case of hybrid and discontinuous dynamics where actions may lead to difficult-to-escape "trap" states, under a given controller. We first learn dynamics for a system without traps from a randomly collected training set (since we do not know what traps will be encountered online). These "nominal" dynamics allow us to perform tasks in scenarios where the dynamics matches the training data, but when unexpected traps arise in execution, we must find a way to adapt our dynamics and control strategy and continue attempting the task. Our approach, Trap-Aware Model Predictive Control (TAMPC), is a two-level hierarchical control algorithm that reasons about traps and non-nominal dynamics to decide between goal-seeking and recovery policies. An important requirement of our method is the ability to recognize nominal dynamics even when we encounter data that is out-of-distribution w.r.t the training data. We achieve this by learning a representation for dynamics that exploits invariance in the nominal environment, thus allowing better generalization. We evaluate our method on simulated planar pushing and peg-in-hole as well as real robot peg-in-hole problems against adaptive control, reinforcement learning, trap-handling baselines, where traps arise due to unexpected obstacles that we only observe through contact. Our results show that our method outperforms the baselines on difficult tasks, and is comparable to prior trap-handling methods on easier tasks.

cs.RO↗

arXiv2020

Chern Insulators and Topological Flat-bands in Magic-angle Twisted Bilayer Graphene

Magic-angle twisted bilayer graphene (MA-TBG) exhibits intriguing quantum phase transitions triggered by enhanced electron-electron interactions when its flat-bands are partially filled. However, the phases themselves and their connection to the putative non-trivial topology of the flat bands are largely unexplored. Here we report transport measurements revealing a succession of doping-induced Lifshitz transitions that are accompanied by van Hove singularities (VHS) which facilitate the emergence of correlation-induced gaps and topologically non-trivial sub-bands. In the presence of a magnetic field, well quantized Hall plateaus at filling of 1, 2, 3 carriers per moiré-cell reveal the sub-band topology and signal the emergence of Chern insulators with Chern-numbers, ! = !, !, !, respectively. Surprisingly, for magnetic fields exceeding 5T we observe a VHS at a filling of 3.5, suggesting the possibility of a fractional Chern insulator. This VHS is accompanied by a crossover from low-temperature metallic, to high-temperature insulating behavior, characteristic of entropically driven Pomeranchuk-like transitions,

cond-mat.mes-hall↗

arXiv2020

A probabilistic approach to the $Φ$-variation of classical fractal functions with critical roughness

We consider Weierstraß and Takagi-van der Waerden functions with critical degree of roughness. In this case, the functions have vanishing $p^{\text{th}}$ variation for all $p>1$ but are also nowhere differentiable and hence not of bounded variation either. We resolve this apparent puzzle by showing that these functions have finite, nonzero, and linear Wiener--Young $Φ$-variation along the sequence of $b$-adic partitions, where $Φ(x)=x/\sqrt{-\log x}$. For the Weierstraß functions, our proof is based on the martingale central limit theorem (CLT). For the Takagi--van der Waerden functions, we use the CLT for Markov chains if a certain parameter $b$ is odd, and the standard CLT for $b$ even.

math.PR↗

arXiv2020

On the $p^{\text{th}}$ variation of a class of fractal functions

The concept of the $p^{\text{th}}$ variation of a continuous function $f$ along a refining sequence of partitions is the key to a pathwise Itô integration theory with integrator $f$. Here, we analyze the $p^{\text{th}}$ variation of a class of fractal functions, containing both the Takagi--van der Waerden and Weierstraß functions. We use a probabilistic argument to show that these functions have linear $p^{\text{th}}$ variation for a parameter $p\ge1$, which can be interpreted as the reciprocal Hurst parameter of the function. It is shown moreover that if functions are constructed from (a skewed version of) the tent map, then the slope of the $p^{\text{th}}$ variation can be computed from the $p^{\text{th}}$ moment of a (non-symmetric) infinite Bernoulli convolution. Finally, we provide a recursive formula of these moments and use it to discuss the existence and non-existence of a signed version of the $p^{\text{th}}$ variation, which occurs in pathwise Itô calculus when $p\ge3$ is an odd integer.

math.PR↗

arXiv2019

Electrostatic Imaging of Encapsulated Graphene

Devices made from two dimensional materials such as graphene and transition metal dichalcogenides exhibit remarkable electronic properties of interest to many subdisciplines of nanoscience. Owing to their 2D nature, their quality is highly susceptible to contamination and degradation when exposed to the ambient environment. Protecting the 2D layers by encapsulation between hexagonal boron nitride layers significantly improves their quality. Locating these samples within the encapsulant and assessing their integrity prior to further processing then becomes challenging. Here we show that conductive scanning probe techniques such as electrostatic force and Kelvin force microscopy makes it possible to visualize the encapsulated layers, their charge environment and local defects including cracks and bubbles on the sub-micrometer scale. Our techniques are employed without requiring electrical contact to the embedded layer, providing valuable feedback on the local electronic quality prior to any device etching or electrode deposition. We show that these measurement modes, which are simple extensions of atomic force microscopy, are perfectly suited for imaging encapsulated conductors and their local charge environments.

cond-mat.mes-hall↗

arXiv2019

Optimal Power Flow in Hybrid AC and Multi-terminal HVDC Networks with Offshore Wind Farm Integration Based on Semidefinite Programming

Multi-terminal high voltage direct current (MTHVDC) technology is a promising technology for the offshore wind farm integration, which requires the new control and operation scheme. Therefore, the optimal power flow problem for this system is important to achieve the optimal economic operation. In this paper, convex relaxation model based on semidefinite programming for the MT-HVDC system considering DC/DC converters is proposed to solve the optimal power flow problem. A hybrid AC and MT-HVDC system for offshore wind farm integration is used for the test. The simulation results validate the effectiveness of the proposed model and guarantee that the global optimum solution is achieved.

eess.SY↗

arXiv2019

On discrete-time self-similar processes with stationary increments

In this paper we study the self-similar processes with stationary increments in a discrete-time setting. Different from the continuous-time case, it is shown that the scaling function of such a process may not take the form of a power function $b(a)=a^H$. More precisely, its scaling function can belong to one of three types, among which one type is degenerate, one type has a continuous-time counterpart, while the other type is new and unique for the discrete-time setting. We then focus on this last type of processes, construct two classes of examples, and prove a special spectral representation result for the processes of this type. We also derive basic properties of discrete-time self-similar processes with stationary increments of different types.

math.PR↗