arXiv subjects

Yifeng Yu

Publications and source records attributed to Yifeng Yu.

At least 37 records · Page 2Linked to original sources

arXiv2025

Does Yakhot's growth law for turbulent burning velocity hold?

Using formal renormalization theory, Yakhot derived in ([32], 1988) an $O\left(\frac{A}{\sqrt{\log A}}\right)$ growth law of the turbulent flame speed with respect to large flow intensity $A$ based on the inviscid G-equation. Although this growth law is widely cited in combustion literature, there has been no rigorous mathematical discussion to date about its validity. As a first step towards unveiling the mystery, we prove that there is no intermediate growth law between $O\left(\frac{A}{\log A}\right)$ and $O(A)$ for two dimensional incompressible Lipschitz continuous periodic flows with bounded swirl sizes. In particular, we do not assume the non-degeneracy of critical points. Additionally, other examples of flows with lower regularity, Lagrangian chaos, and related phenomena are also discussed.

math.AP↗

arXiv2024

VISinger2+: End-to-End Singing Voice Synthesis Augmented by Self-Supervised Learning Representation

Singing Voice Synthesis (SVS) has witnessed significant advancements with the advent of deep learning techniques. However, a significant challenge in SVS is the scarcity of labeled singing voice data, which limits the effectiveness of supervised learning methods. In response to this challenge, this paper introduces a novel approach to enhance the quality of SVS by leveraging unlabeled data from pre-trained self-supervised learning models. Building upon the existing VISinger2 framework, this study integrates additional spectral feature information into the system to enhance its performance. The integration aims to harness the rich acoustic features from the pre-trained models, thereby enriching the synthesis and yielding a more natural and expressive singing voice. Experimental results in various corpora demonstrate the efficacy of this approach in improving the overall quality of synthesized singing voices in both objective and subjective metrics.

cs.SD↗

arXiv2024

Muskits-ESPnet: A Comprehensive Toolkit for Singing Voice Synthesis in New Paradigm

This research presents Muskits-ESPnet, a versatile toolkit that introduces new paradigms to Singing Voice Synthesis (SVS) through the application of pretrained audio models in both continuous and discrete approaches. Specifically, we explore discrete representations derived from SSL models and audio codecs and offer significant advantages in versatility and intelligence, supporting multi-format inputs and adaptable data processing workflows for various SVS models. The toolkit features automatic music score error detection and correction, as well as a perception auto-evaluation module to imitate human subjective evaluating scores. Muskits-ESPnet is available at \url{https://github.com/espnet/espnet}.

cs.SD↗

arXiv2024

Bifurcation of homogenization and nonhomogenization of the curvature G-equation with shear flows

The level-set curvature G-equation, a well-known model in turbulent combustion, has the following form $G_t + \left(1-d\, \mathrm{dvi}\left({\frac{DG}{|DG|}}\right)\right)_+|DG|+V(X)\cdot DG=0.$ Here the cutoff correction $()_+$ is imposed to avoid non-physical negative local burning velocity. The existence of the effective burning velocity has been established for a large class of physically relevant incompressible flows $V$ in two dimensions [13] via game theory dynamics. In this paper, we show that the effective burning velocity associated with shear flows in dimensions three or higher ceases to exist when the flow intensity surpasses a bifurcation point. The characterization of the bifurcation point in three dimensions is closely related to the regularity theory of two-dimensional minimal surface type equations due to [29]. As a consequence, a bifurcation also exists for the validity of full homogenization of the curvature G-equation associated with shear flows.

math.AP↗

arXiv2024

Singing Voice Data Scaling-up: An Introduction to ACE-Opencpop and ACE-KiSing

In singing voice synthesis (SVS), generating singing voices from musical scores faces challenges due to limited data availability. This study proposes a unique strategy to address the data scarcity in SVS. We employ an existing singing voice synthesizer for data augmentation, complemented by detailed manual tuning, an approach not previously explored in data curation, to reduce instances of unnatural voice synthesis. This innovative method has led to the creation of two expansive singing voice datasets, ACE-Opencpop and ACE-KiSing, which are instrumental for large-scale, multi-singer voice synthesis. Through thorough experimentation, we establish that these datasets not only serve as new benchmarks for SVS but also enhance SVS performance on other singing voice datasets when used as supplementary resources. The corpora, pre-trained models, and their related training recipes are publicly available at ESPnet-Muskits (\url{https://github.com/espnet/espnet})

cs.SD↗

arXiv2024

Global Well-posedness and Convergence Analysis of Score-based Generative Models via Sharp Lipschitz Estimates

We establish global well-posedness and convergence of the score-based generative models (SGM) under minimal general assumptions of initial data for score estimation. For the smooth case, we start from a Lipschitz bound of the score function with optimal time length. The optimality is validated by an example whose Lipschitz constant of scores is bounded at initial but blows up in finite time. This necessitates the separation of time scales in conventional bounds for non-log-concave distributions. In contrast, our follow up analysis only relies on a local Lipschitz condition and is valid globally in time. This leads to the convergence of numerical scheme without time separation. For the non-smooth case, we show that the optimal Lipschitz bound is O(1/t) in the point-wise sense for distributions supported on a compact, smooth and low-dimensional manifold with boundary.

cs.LG↗

arXiv2024

Optimal rate of convergence in periodic homogenization of viscous Hamilton-Jacobi equations

We study the optimal rate of convergence in periodic homogenization of the viscous Hamilton-Jacobi equation $u^\varepsilon_t + H(\frac{x}{\varepsilon},Du^\varepsilon) = \varepsilon Δu^\varepsilon$ in $\mathbb R^n\times (0,\infty)$ subject to a given initial datum. We prove that $\|u^\varepsilon-u\|_{L^\infty(\mathbb R^n \times [0,T])} \leq C(1+T) \sqrt{\varepsilon}$ for any given $T>0$, where $u$ is the viscosity solution of the effective problem. Moreover, we show that the $O(\sqrt{\varepsilon})$ rate is optimal for a natural class of $H$ and a Lipschitz continuous initial datum, both theoretically and through numerical experiments. It remains an interesting question to investigate whether the convergence rate can be improved when $H$ is uniformly convex. Finally, we propose a numerical scheme for the approximation of the effective Hamiltonian based on a finite element approximation of approximate corrector problems.

math.AP↗

arXiv2024

Lagrangian, Game Theoretic and PDE Methods for Averaging G-equations in Turbulent Combustion: Existence and Beyond

G-equations are popular level set Hamilton-Jacobi nonlinear partial differential equations (PDEs) of first or second order arising in turbulent combustion. Characterizing the effective burning velocity (also known as the turbulent burning velocity) is a fundamental problem there. We review relevant studies of the G-equation models with a focus on both the existence of effective burning velocity (homogenization), and its dependence on physical and geometric parameters (flow intensity and curvature effect) through representative examples. The corresponding physical background is also presented to provide motivations for mathematical problems of interest. The lack of coercivity of Hamiltonian is a hallmark of G-equations. When either the curvature of the level set or the strain effect of fluid flows is accounted for, the Hamiltonian becomes highly non-convex and nonlinear. In the absence of coercivity and convexity, PDE (Eulerian) approach suffers from insufficient compactness to establish averaging (homogenization). We review and illustrate a suite of Lagrangian tools, most notably min-max (max-min) game representations of curvature and strain G-equations, working in tandem with analysis of streamline structures of fluid flows and PDEs. We discuss open problems for future development in this emerging area of dynamic game analysis for averaging non-coercive, non-convex, and nonlinear PDEs such as geometric (curvature-dependent) PDEs with advection.

math.AP↗

arXiv2023

Existence of an effective burning velocity in cellular flow for curvature G-equation via game analysis

G-equation is a popular level set model in turbulent combustion, and becomes an advective mean curvature type evolution equation when curvature of a moving flame in a fluid flow is considered: $$ G_t + \left(1-d\, \mathrm{Div}{\frac{DG}{|DG|}}\right)_+|DG|+V(x)\cdot DG=0. $$ Here $d>0$ is the Markstein number and the positive part $()_+$ is imposed to avoid a non-physical negative laminar flame speed. For simplicity of presentation, we focus mainly on the case when $V:\mathbb{R}^2\to \mathbb{R}^2$ is the two dimensional cellular flow with Hamiltonian $H = \sin x_1 \, \sin x_2$ and amplitude $A$. Our main result is that for any unit vector $p\in \mathbb{R}^2$, there exists a positive number $\overline H(p)$ such that if $G(x,0)=p\cdot x$, then $$ \left|G(x,t)-p\cdot x+\overline H(p)t\right|\leq C \quad \text{in $\mathbb{R}^2\times [0,\infty)$} $$ for a constant $C$ depending only on the Markstein number $d$ and the cellular flow amplitude $A$. The number $\overline H(p)$ corresponds to the effective burning velocity in the physics literature. The non-coercivity encountered here is one of the major difficulties for homogenization of the mean curvature-type equations. To overcome it, we introduce a new approach that combines PDE methods with a dynamical analysis of the Kohn-Serfaty deterministic game characterization of the curvature G-equation utilizing the streamline structure of cellular flows. Extension to general two-dimensional incompressible flows is also discussed. In three dimensional incompressible flows, the existence of $\overline H(p)$ might fail when the flow intensity exceeds a bifurcation value even for simple shear flows [32].

math.AP↗

arXiv2023

A Systematic Exploration of Joint-training for Singing Voice Synthesis

There has been a growing interest in using end-to-end acoustic models for singing voice synthesis (SVS). Typically, these models require an additional vocoder to transform the generated acoustic features into the final waveform. However, since the acoustic model and the vocoder are not jointly optimized, a gap can exist between the two models, leading to suboptimal performance. Although a similar problem has been addressed in the TTS systems by joint-training or by replacing acoustic features with a latent representation, adopting corresponding approaches to SVS is not an easy task. How to improve the joint-training of SVS systems has not been well explored. In this paper, we conduct a systematic investigation of how to better perform a joint-training of an acoustic model and a vocoder for SVS. We carry out extensive experiments and demonstrate that our joint-training strategy outperforms baselines, achieving more stable performance across different datasets while also increasing the interpretability of the entire framework.

cs.SD↗

arXiv2022

Optimal convergence rate for periodic homogenization of convex Hamilton-Jacobi equations

In this paper, we show that the rate of convergence in periodic homogenization of convex Hamilton-Jacobi equations is always $O(\varepsilon)$, which is optimal. This is a natural extension of a result concerning stable norms in metric geometry [4] that is essentially equivalent to the homogenization of convex static Hamilton-Jacobi equations. Another extremely interesting question in this direction is whether the $O(\varepsilon)$ rate holds in the nonconvex setting. We present a special nonconvex example with $O(\varepsilon)$ convergence rate, which relies on identifying the shape of the effective Hamiltonian and game theory interpretation formulas.

math.AP↗

arXiv2022

Differentiability of effective fronts in the continuous setting in two dimensions

We study the effective front associated with first-order front propagations in two dimensions ($n=2$) in the periodic setting with continuous coefficients. Our main result says that that the boundary of the effective front is differentiable at every irrational point. Equivalently, the stable norm associated with a continuous $\mathbb{Z}^2$-periodic Riemannian metric is differentiable at irrational points. This conclusion was obtained decades ago for smooth metrics ([3,5]). To the best of our knowledge, our result provides the first nontrivial property of the effective fronts in the continuous setting, which is the standard assumption in the PDE theory. Combining with the sufficiency result in [12], our result implies that for continuous coefficients, a polygon could be an effective front if and only if it is centrally symmetric with rational vertices and nonempty interior.

math.AP↗

arXiv2022

Remarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form

We study and characterize the optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form. We obtain that the optimal rate of convergence is either $O(\varepsilon)$ or $O(\varepsilon^2)$ depending on the diffusion matrix $A$, source term $f$, and boundary data $g$. Moreover, we show that the set of diffusion matrices $A$ that give optimal rate $O(\varepsilon)$ is open and dense in the set of $C^{2,α}$ periodic, symmetric, and positive definite matrices, which means that generically, the optimal rate is $O(\varepsilon)$.

math.AP↗

arXiv2021

Effective fronts of polygon shapes in two dimensions

We study the effective fronts of first order front propagations in two dimensions ($n=2$) in the periodic setting. Using PDE-based approaches, we show that for every $α\in (0,1)$, the class of centrally symmetric polygons with rational vertices and nonempty interior is admissible as effective fronts for given front speeds in $C^{1,α}(\mathbb T^2,(0,\infty))$. This result can also be formulated in the language of stable norms corresponding to periodic metrics in $\mathbb T^2$. Similar results were known long time ago when $n\geq 3$ for front speeds in $C^{\infty}(\mathbb T^n,(0,\infty))$. Due to topological restrictions, the two dimensional case is much more subtle. In fact, the effective front is $C^1$, which cannot be a polygon, for given $C^{1,1}(\mathbb T^2,(0,\infty))$ front speeds. Our regularity requirements on front speeds are hence optimal. To the best of our knowledge, this is the first time that polygonal effective fronts have been constructed in two dimensions.

math.AP↗

arXiv2020

High Degeneracy of Effective Hamiltonian in Two Dimensions

Consider the effective Hamiltonian $\overline H(p)$ associated with the mechanical Hamiltonian $H(p,x)={1\over 2}|p|^2+V(x)$. We prove that for generic $V$, $\overline H$ is piecewise 1d in a dense open set in two dimensions using Aubry-Mather theory.

math.AP↗

arXiv2019

Computing Residual Diffusivity by Adaptive Basis Learning via Super-Resolution Deep Neural Networks

It is expensive to compute residual diffusivity in chaotic in-compressible flows by solving advection-diffusion equation due to the formation of sharp internal layers in the advection dominated regime. Proper orthogonal decomposition (POD) is a classical method to construct a small number of adaptive orthogonal basis vectors for low cost computation based on snapshots of fully resolved solutions at a particular molecular diffusivity $D_{0}^{*}$. The quality of POD basis deteriorates if it is applied to $D_0\ll D_{0}^{*}$. To improve POD, we adapt a super-resolution generative adversarial deep neural network (SRGAN) to train a nonlinear mapping based on snapshot data at two values of $D_{0}^{*}$. The mapping models the sharpening effect on internal layers as $D_0$ becomes smaller. We show through numerical experiments that after applying such a mapping to snapshots, the prediction accuracy of residual diffusivity improves considerably that of the standard POD.

physics.comp-ph↗

arXiv2019

Effective fronts of polytope shapes

We study the periodic homogenization of first order front propagations. Based on PDE methods, we provide a simple proof that for $n \geq 3$, the class of centrally symmetric polytopes with rational coordinates and nonempty interior is admissible as effective fronts, which was also established in [1,10] in the form of stable norms as an extension of Hedlund's classical result [7]. Besides, we obtain the optimal convergence rate of the homogenization problem for this class.

math.AP↗

arXiv2019

Rate of convergence in periodic homogenization of Hamilton-Jacobi equations: the convex setting

We study the rate of convergence of $u^ε$, as $ε\to 0+$, to $u$ in periodic homogenization of Hamilton-Jacobi equations. Here, $u^ε$ and $u$ are viscosity solutions to the oscillatory Hamilton-Jacobi equation and its effective equation \begin{equation*} {\rm (C)_ε} \qquad \begin{cases} u_t^ε+H\left(\frac{x}ε,Du^ε\right)=0 \qquad &\text{in} \ \mathbb{R}^n \times (0,\infty), u^ε(x,0)=g(x) \qquad &\text{on} \ \mathbb{R}^n, \end{cases} \end{equation*} and \begin{equation*} {\rm (C)} \qquad \begin{cases} u_t+\overline{H}\left(Du\right)=0 \qquad &\text{in} \ \mathbb{R}^n \times (0,\infty), u(x,0)=g(x) \qquad &\text{on} \ \mathbb{R}^n, \end{cases} \end{equation*} respectively. We assume that the Hamiltonian $H=H(y,p)$ is coercive and convex in the $p$ variable and is $\mathbb{Z}^n$-periodic in the $y$ variable, and the initial data $g$ is bounded and Lipschitz continuous.

math.AP↗