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Jiahui Hong

Publications and source records attributed to Jiahui Hong.

12 recordsLinked to original sources
arXiv2026

Recent progress in generalized Hamiltonian gradient flow: Singularities

This paper is a survey of the generalized Hamiltonian gradient flow (GHGF) framework for Hamilton-Jacobi equations, with an emphasis on the propagation of singularities and its connections to weak KAM theory, optimal transport and mean field control. In addition to reviewing the main ideas and known results, we present two new contributions. First, we provide a variational construction of generalized characteristics via a minimizing movement scheme; by taking the weak limit of approximate solutions and using Young measure compactness, we show that the limiting curve satisfies the generalized characteristic differential inclusion. Second, we lift the forward dynamics of strict singular characteristics to a semi-flow on the space of trajectories and study its invariant probability measures. We prove that the only invariant measures of the GHGF semi-flow that attain the critical value \(c[H]\) are precisely the projected Mather measures, thereby giving a new dynamical characterization of Mather's minimal measures as well as Ma\~n\'e's critical value. Finally, we discuss a number of open problems that arise from the GHGF perspective, including questions on uniqueness of strict singular characteristics, rectifiability of cut loci, stability under perturbations, contact Hamiltonian systems, vanishing noise limits, and extensions to non-convex or low-regularity Hamiltonians. These problems highlight the deeper connections between singular dynamics, ergodic theory, optimal transport, and geometric analysis, and indicate directions for future research.

math.AP↗
arXiv2025

Bidirectional Time-Frequency Pyramid Network for Enhanced Robust EEG Classification

Existing EEG recognition models suffer from poor cross-paradigm generalization due to dataset-specific constraints and individual variability. To overcome these limitations, we propose BITE (Bidirectional Time-Freq Pyramid Network), an end-to-end unified architecture featuring robust multistream synergy, pyramid time-frequency attention (PTFA), and bidirectional adaptive convolutions. The framework uniquely integrates: 1) Aligned time-frequency streams maintaining temporal synchronization with STFT for bidirectional modeling, 2) PTFA-based multi-scale feature enhancement amplifying critical neural patterns, 3) BiTCN with learnable fusion capturing forward/backward neural dynamics. Demonstrating enhanced robustness, BITE achieves state-of-the-art performance across four divergent paradigms (BCICIV-2A/2B, HGD, SD-SSVEP), excelling in both within-subject accuracy and cross-subject generalization. As a unified architecture, it combines robust performance across both MI and SSVEP tasks with exceptional computational efficiency. Our work validates that paradigm-aligned spectral-temporal processing is essential for reliable BCI systems. Just as its name suggests, BITE "takes a bite out of EEG." The source code is available at https://github.com/cindy-hong/BiteEEG.

cs.LG↗
arXiv2025

Global propagation of singularities for magnetic mechanical systems

We prove that singularities propagate globally for viscosity solutions of Hamilton-Jacobi equations related to magnetic mechanical systems on closed Riemannian manifolds. Our main result shows that for any weak KAM solution $u$, the singular set $\text{Sing}\,(u)$ remains invariant under the generalized gradient flow dynamics. The proof combines three key elements: (1) reduction from magnetic to Riemannian systems, (2) analysis of reparameterized flows, and (3) regularization techniques. Compared to previous analytic approaches, our geometric method provides clearer insights into the underlying Riemannian structure. We also establish necessary conditions for singularity existence, particularly when the Euler characteristic is nonzero and the magnetic form is non-exact. This approach does not extend directly to Finsler metrics due to structural differences.

math.AP↗
arXiv2024

Variational construction of singular characteristics and propagation of singularities

On a smooth closed manifold $M$, we introduce a novel theory of maximal slope curves for any pair $(\phi,H)$ with $\phi$ a semiconcave function and $H$ a Hamiltonian. By using the notion of maximal slope curve from gradient flow theory, the intrinsic singular characteristics constructed in [Cannarsa, P.; Cheng, W., \textit{Generalized characteristics and Lax-Oleinik operators: global theory}. Calc. Var. Partial Differential Equations 56 (2017), no. 5, 56:12], the smooth approximation method developed in [Cannarsa, P.; Yu, Y. \textit{Singular dynamics for semiconcave functions}. J. Eur. Math. Soc. 11 (2009), no. 5, 999--1024], and the broken characteristics studied in [Khanin, K.; Sobolevski, A., \textit{On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations}. Arch. Ration. Mech. Anal. 219 (2016), no. 2, 861--885], we prove the existence and stability of such maximal slope curves and discuss certain new weak KAM features. We also prove that maximal slope curves for any pair $(\phi,H)$ are exactly broken characteristics which have right derivatives everywhere. Applying this theory, we establish a global variational construction of strict singular characteristics and broken characteristics. Moreover, we prove a result on the global propagation of cut points along generalized characteristics, as well as a result on the propagation of singular points along strict singular characteristics, for weak KAM solutions. We also obtain the continuity equation along strict singular characteristics which clarifies the mass transport nature in the problem of propagation of singularities.

math.AP↗
arXiv2024

Optimal transport in the frame of abstract Lax-Oleinik operator revisited

This is our first paper on the extension of our recent work on the Lax-Oleinik commutators and its applications to the intrinsic approach of propagation of singularities of the viscosity solutions of Hamilton-Jacobi equations. We reformulate Kantorovich-Rubinstein duality theorem in the theory of optimal transport in terms of abstract Lax-Oleinik operators, and analyze the relevant optimal transport problem in the case the cost function $c(x,y)=h(t_1,t_2,x,y)$ is the fundamental solution of Hamilton-Jacobi equation. For further applications to the problem of cut locus and propagation of singularities in optimal transport, we introduce corresponding random Lax-Oleinik operators. We also study the problem of singularities for $c$-concave functions and its dynamical implication when $c$ is the fundamental solution with $t_2-t_1\ll1$ and $t_2-t_1<\infty$, and $c$ is the Peierls' barrier respectively.

math.AP↗
arXiv2023

Topological and control theoretic properties of Hamilton-Jacobi equations via Lax-Oleinik commutators

In the context of weak KAM theory, we discuss the commutators $\{T^-_t\circ T^+_t\}_{t\geqslant0}$ and $\{T^+_t\circ T^-_t\}_{t\geqslant0}$ of Lax-Oleinik operators. We characterize the relation $T^-_t\circ T^+_t=Id$ for both small time and arbitrary time $t$. We show this relation characterizes controllability for evolutionary Hamilton-Jacobi equation. Based on our previous work on the cut locus of viscosity solution, we refine our analysis of the cut time function $\tau$ in terms of commutators $T^+_t\circ T^-_t-T^+_t\circ T^-_t$ and clarify the structure of the super/sub-level set of the cut time function $\tau$.

math.AP↗
arXiv2022

Topology of singular set of semiconcave function via Arnaud's theorem

We proved the (local) path-connectedness of certain subset of the singular set of semiconcave functions with linear modulus in general. In some sense this result is optimal. The proof is based on a theorem by Marie-Claude Arnaud (M.-C. Arnaud, \textit{Pseudographs and the Lax-Oleinik semi-group: a geometric and dynamical interpretation}. Nonlinearity, \textbf{24}(1): 71-78, 2011.). We also gave a new proof of the theorem in time-dependent case.

math.AP↗
arXiv2022

Local strict singular characteristics II: existence for stationary equation on $\mathbb{R}^2$

The notion of strict singular characteristics is important in the wellposedness issue of singular dynamics on the cut locus of the viscosity solutions. We provide an intuitive and rigorous proof of the existence of the strict singular characteristics of Hamilton-Jacobi equation $H(x,Du(x),u(x))=0$ in two dimensional case. We also proved if $\mathbf{x}$ is a strict singular characteristic, then we really have the right-differentiability of $\mathbf{x}$ and the right-continuity of $\dot{\mathbf{x}}^+(t)$ for every $t$. Such a strict singular characteristic must give a selection $p(t)\in D^+u(\mathbf{x}(t))$ such that $p(t)=\arg\min_{p\in D^+u(\mathbf{x}(t))}H(\mathbf{x}(t),p,u(\mathbf{x}(t)))$.

math.AP↗
arXiv2021

Global propagation of singularities for discounted Hamilton-Jacobi equations

The main purpose of this paper is to study the global propagation of singularities of viscosity solution to discounted Hamilton-Jacobi equation \begin{equation}\label{eq:discount 1}\tag{HJ$_\lambda$} \lambda v(x)+H( x, Dv(x) )=0 , \quad x\in \mathbb{R}^n. \end{equation} We reduce the problem for equation \eqref{eq:discount 1} into that for a time-dependent evolutionary Hamilton-Jacobi equation. We proved that the singularities of the viscosity solution of \eqref{eq:discount 1} propagate along locally Lipschitz singular characteristics which can extend to $+\infty$. We also obtained the homotopy equivalence between the singular set and the complement of associated the Aubry set with respect to the viscosity solution of equation \eqref{eq:discount 1}.

math.DS↗
arXiv2021

Local strict singular characteristics: Cauchy problem with smooth initial data

Main purpose of this paper is to study the local propagation of singularities of viscosity solution to contact type evolutionary Hamilton-Jacobi equation $$ D_tu(t,x)+H(t,x,D_xu(t,x),u(t,x))=0. $$ An important issue of this topic is the existence, uniqueness and regularity of the strict singular characteristic. We apply the recent existence and regularity results on the Herglotz' type variational problem to the aforementioned Hamilton-Jacobi equation with smooth initial data. We obtain some new results on the local structure of the cut set of the viscosity solution near non-conjugate singular points. Especially, we obtain an existence result of smooth strict singular characteristic from and to non-conjugate singular initial point based on the structure of the superdifferential of the solution, which is even new in the classical time-dependent case. We also get a global propagation result for the $C^1$ singular support in the contact case.

math.AP↗