arXiv subjects

Kuang Huang

Publications and source records attributed to Kuang Huang.

At least 19 recordsLinked to original sources

arXiv2026

Efficient Weak-Entropy PINN for Solving Hyperbolic Conservation Laws

In recent years, neural networks have significantly advanced numerical solutions of partial differential equations (PDEs). However, solving PDEs with discontinuous solutions, such as hyperbolic conservation laws, remains challenging for neural network-based methods such as physics-informed neural networks (PINNs). Existing methods often rely on strong prior assumptions such as knowledge of discontinuity locations, or they introduce artificial smoothing terms that degrade accuracy. However, accurately solving these conservation laws and predicting the formation and propagation of discontinuities in solutions is crucial in many practical applications, including gas dynamics and traffic flow modeling. In this paper, we introduce a novel Weak-Entropy PINN (WEPINN) framework for hyperbolic conservation laws with discontinuous solutions. The method enforces the governing equations in their weak (integral) formulation and incorporates the entropy condition to select the physically admissible solution, while employing the discrete fast Fourier transform (DFFT) for efficient numerical integration. Our method is tested through extensive numerical experiments on a variety of scalar conservation laws and systems of conservation laws in one and two dimensional spaces. These experiments demonstrate that our method can accurately resolve sharp discontinuities while effectively capturing interactions between multiple shock and rarefaction waves.

math.NA↗

arXiv2026

Stability Estimates for the Inverse Problem of Reconstructing Point sources in Parabolic Equations

In this work, we investigate the stability issue of the inverse problem of determining the locations and time-dependent amplitudes of point sources in a parabolic equation with a non-self adjoint elliptic operator from boundary observations. We derive different stability estimates for determining the locations and the amplitudes of the sources in the space, the plane as well as in dimension one. The analysis employs a novel approach that combines several different arguments, including the improved regularity of the solutions, the application of Carleman estimates, time extension of solutions, and construction of explicit solutions to the adjoint equations. Further we provide numerical reconstructions to complement the theoretical findings.

math.AP↗

arXiv2025

A Volterra equation approach to the local limit of nonlocal traffic models

We consider a class of nonlocal conservation laws modeling traffic flow, given by $ \partial_t u_\varepsilon + \partial_x(V(u_\varepsilon \ast \gamma_\varepsilon)\, u_\varepsilon) = 0 $ with $ \gamma_\varepsilon(\cdot) := \varepsilon^{-1}\gamma(\cdot/\varepsilon) $ for a suitable convex convolution kernel $\gamma$. Since the work of Colombo et al. (Arch. Ration. Mech. Anal., 2023), thanks to uniform $ \mathrm{L}^\infty $- and TV-estimates, it is known that $ w_\varepsilon := u_\varepsilon \ast \gamma_\varepsilon $ converges to the entropy solution of the local scalar conservation law $ \partial_t u + \partial_x(V(u)\, u) = 0 $ as $\varepsilon \searrow 0$. However, the convergence of $ \{u_\varepsilon\}_{\varepsilon > 0} $ itself has not been fully addressed so far. In this direction, a known result applies specifically to the case of an exponential kernel, where the identity $ \varepsilon \partial_x w_\varepsilon = w_\varepsilon - u_\varepsilon $ is fundamental. In this work, we address this gap in the literature and prove that $ \{u_\varepsilon\}_{\varepsilon > 0} $ converges to the same limit $u$ under the mild additional assumption that the initial datum belongs to $\mathrm L^1(\mathbb{R})$. Our analysis exploits, through a Fourier approach, the stability properties of the more general Volterra-type equation $\partial_xw_\varepsilon=\gamma'_\varepsilon\ast u_\varepsilon-\gamma_\varepsilon(0)u_\varepsilon$, thereby deducing the convergence of $u_\varepsilon$ from that of $w_\varepsilon$.

math.AP↗

arXiv2025

Singular limit for a class of nonlocal conservation laws via compensated compactness

We consider a class of nonlocal conservation laws modeling traffic flows, given by $ \partial_t u_\varepsilon + \partial_x(V(u_\varepsilon \ast \gamma_\varepsilon) u_\varepsilon) = 0$, with a rescaled convolution kernel $\gamma_\varepsilon(\cdot) := \varepsilon^{-1}\gamma(\cdot/\varepsilon)$. We establish the strong $\mathrm L^1_{\mathrm{loc}}$-convergence of weak solutions $u_\varepsilon$ toward the entropy-admissible solution of the corresponding local conservation law as the kernel $\gamma_\varepsilon$ concentrates to a Dirac delta distribution when $\varepsilon \searrow 0$. In contrast to previous literature, we obtain compactness of the family $\{u_\varepsilon \ast \gamma_\varepsilon\}_{\varepsilon>0}$ without relying on total variation bounds or Ole\u{\i}nik-type estimates. Instead, we establish $\mathrm L^2$-type bounds on its entropy production and use the theory of compensated compactness, assuming that the initial datum merely belongs to $\mathrm L^1\cap \mathrm L^\infty$. Our results are twofold. First, we establish the nonlocal-to-local limit for the piecewise constant kernel $\gamma(\cdot) := {1}_{[-1,0]}(\cdot)$ combined with the affine velocity function from Greenshields' traffic model. Second, we prove the limit for strictly monotone kernels along with decreasing velocity functions. These results settle a long-standing open problem concerning the nonlocal-to-local convergence for non-convex kernels.

math.AP↗

arXiv2025

Asymptotically compatible entropy-consistent discretization for a class of nonlocal conservation laws

We consider a class of nonlocal conservation laws modeling traffic flows, given by $ \partial_t \rho_\varepsilon + \partial_x(V(\rho_\varepsilon \ast \gamma_\varepsilon) \rho_\varepsilon) = 0 $ with a suitable convex kernel $ \gamma_\varepsilon $, and its Godunov-type numerical discretization. We prove that, as the nonlocal parameter $ \varepsilon $ and mesh size $ h $ tend to zero simultaneously, the discrete approximation $ W_{\varepsilon,h} $ of $ W_\varepsilon := \rho_\varepsilon \ast \gamma_\varepsilon $ converges to the entropy solution of the (local) scalar conservation law $ \partial_t \rho + \partial_x(V(\rho) \rho) = 0 $, with an explicit convergence rate estimate of order $ \varepsilon+h+\sqrt{\varepsilon\, t}+\sqrt{h\,t} $. In particular, with an exponential kernel, we establish the same convergence result for the discrete approximation $ \rho_{\varepsilon,h} $ of $ \rho_\varepsilon $, along with an $ \mathrm{L}^1 $-contraction property for $ W_\varepsilon $. The key ingredients in proving these results are uniform $ \mathrm{L}^\infty $- and $\mathrm{TV}$-estimates that ensure compactness of approximate solutions, and discrete entropy inequalities that ensure the entropy admissibility of the limit solution.

math.NA↗

arXiv2025

H\"older Stable Recovery of the Source in Space-Time Fractional Wave Equations

We study the recovery of a spatially dependent source in a one-dimensional space-time fractional wave equation using boundary measurement data collected at a single endpoint. The main challenge arises from the fact that the eigenfunctions of the Dirichlet eigenvalue problem do not form an orthogonal system, due to the presence of a fractional derivative in space. To address this difficulty, we introduce a bi-orthogonal basis for the Mittag-Leffler functions and use it to establish uniqueness and H\"older-type stability results, provided the measurement time is sufficiently large. A Tikhonov regularization method is then employed to numerically solve the inverse source problem. Several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method and to validate our theoretical findings.

math.AP↗

arXiv2025

Homogenization rates of beam lattices to micropolar continua

As the size of a mechanical lattice with beam-modeled edges approaches zero, it undergoes homogenization into a continuum model, which exhibits unusual mechanical properties that deviate from classical Cauchy elasticity, named micropolar elasticity. Typically, the homogenization process is qualitative in the engineering community, lacking quantitative homogenization error estimates. In this paper, we rigorously analyze the homogenization process of a beam lattice to a continuum. Our approach is initiated from an engineered mechanical problem defined on a triangular lattice with periodic boundary conditions. By applying Fourier transformations, we reduce the problem to a series of equations in the frequency domain. As the lattice size approaches zero, this yields a homogenized model in the form of a partial differential equation with periodic boundary conditions. This process can be easily justified if the external conditions in the frequency domain are nonzero only at low-frequency modes. However, through numerical experiments, we discover that beyond the low-frequency regime, the homogenization of the beam lattice differs from classical periodic homogenization theory due to the additional rotational degrees of freedom in the beams. A crucial technique in our analysis is the decoupling of displacement and rotation fields, achieved through a linear algebraic manipulation known as the Schur complement. Through dedicated analysis, we establish the coercivity of the Schur complements in both lattice and continuum models, which enables us to derive convergence rate estimates for homogenization errors. Numerical experiments validate the optimality of the homogenization rate estimates.

math.NA↗

arXiv2024

Point Source Identification in Subdiffusion from A Posteriori Internal Measurement

In this work we investigate an inverse problem of recovering point sources and their time-dependent strengths from {a posteriori} partial internal measurements in a subdiffusion model which involves a Caputo fractional derivative in time and a general second-order elliptic operator in space. We establish the well-posedness of the direct problem in the sense of transposition and improved local regularity. Using classical unique continuation of the subdiffusion model and improved local solution regularity, we prove the uniqueness of simultaneously recovering the locations of point sources, time-dependent strengths and initial condition for both one- and multi-dimensional cases. Moreover, in the one-dimensional case, the elliptic operator can have time-dependent coefficients. These results extend existing studies on point source identification for parabolic type problems. Additionally we present several numerical experiments to show the feasibility of numerical reconstruction.

math.AP↗

arXiv2024

Automated Discovery of Operable Dynamics from Videos

Dynamical systems form the foundation of scientific discovery, traditionally modeled with predefined state variables such as the angle and angular velocity, and differential equations such as the equation of motion for a single pendulum. We introduce a framework that automatically discovers a low-dimensional and operable representation of system dynamics, including a set of compact state variables that preserve the smoothness of the system dynamics and a differentiable vector field, directly from video without requiring prior domain-specific knowledge. The prominence and effectiveness of the proposed approach are demonstrated through both quantitative and qualitative analyses of a range of dynamical systems, including the identification of stable equilibria, the prediction of natural frequencies, and the detection of chaotic and limit cycle behaviors. The results highlight the potential of our data-driven approach to advance automated scientific discovery.

eess.SY↗

arXiv2024

On an Inverse Problem of the Generalized Bathtub Model of Network Trip Flows

In this work, we investigate the generalized bathtub model, a nonlocal transport equation for describing network trip flows served by privately operated vehicles inside a road network. First, we establish the well-posedness of the mathematical model for both classical and weak solutions. Then we consider an inverse source problem of the model with model parameters embodying particular traffic situations. We establish a conditional Lipschitz stability of the inverse problem under suitable a priori regularity assumption on the problem data, using a Volterra integral formulation of the problem. Inspired by the analysis, we develop an easy-to-implement numerical method for reconstructing the flow rates, and provide the error analysis of the method. Further we present several numerical experiments to complement the theoretical analysis.

math.AP↗

arXiv2023

Asymptotically compatibility of a class of numerical schemes for a nonlocal traffic flow model

This paper considers numerical discretization of a nonlocal conservation law modeling vehicular traffic flows involving nonlocal inter-vehicle interactions. The nonlocal model involves an integral over the range measured by a horizon parameter and it recovers the local Lighthill-Richards-Whitham model as the nonlocal horizon parameter goes to zero. Good numerical schemes for simulating these parameterized nonlocal traffic flow models should be robust with respect to the change of the model parameters but this has not been systematically investigated in the literature. We fill this gap through a careful study of a class of finite volume numerical schemes with suitable discretizations of the nonlocal integral, which include several schemes proposed in the literature and their variants. Our main contributions are to demonstrate the asymptotically compatibility of the schemes, which includes both the uniform convergence of the numerical solutions to the unique solution of nonlocal continuum model for a given positive horizon parameter and the convergence to the unique entropy solution of the local model as the mesh size and the nonlocal horizon parameter go to zero simultaneously. It is shown that with the asymptotically compatibility, the schemes can provide robust numerical computation under the changes of the nonlocal horizon parameter.

math.NA↗

arXiv2022

A Space-time Nonlocal Traffic Flow Model: Relaxation Representation and Local Limit

We propose and study a nonlocal conservation law modelling traffic flow in the existence of inter-vehicle communication. It is assumed that the nonlocal information travels at a finite speed and the model involves a space-time nonlocal integral of weighted traffic density. The well-posedness of the model is established under suitable conditions on the model parameters and by a suitably-defined initial condition. In a special case where the weight kernel in the nonlocal integral is an exponential function, the nonlocal model can be reformulated as a $2\times2$ hyperbolic system with relaxation. With the help of this relaxation representation, we show that the Lighthill-Whitham-Richards model is recovered in the equilibrium approximation limit.

math.AP↗

arXiv2021

Discovering State Variables Hidden in Experimental Data

All physical laws are described as relationships between state variables that give a complete and non-redundant description of the relevant system dynamics. However, despite the prevalence of computing power and AI, the process of identifying the hidden state variables themselves has resisted automation. Most data-driven methods for modeling physical phenomena still assume that observed data streams already correspond to relevant state variables. A key challenge is to identify the possible sets of state variables from scratch, given only high-dimensional observational data. Here we propose a new principle for determining how many state variables an observed system is likely to have, and what these variables might be, directly from video streams. We demonstrate the effectiveness of this approach using video recordings of a variety of physical dynamical systems, ranging from elastic double pendulums to fire flames. Without any prior knowledge of the underlying physics, our algorithm discovers the intrinsic dimension of the observed dynamics and identifies candidate sets of state variables. We suggest that this approach could help catalyze the understanding, prediction and control of increasingly complex systems. Project website is at: https://www.cs.columbia.edu/~bchen/neural-state-variables

math.DS↗

arXiv2021

A Physics-Informed Deep Learning Paradigm for Traffic State and Fundamental Diagram Estimation

Traffic state estimation (TSE) bifurcates into two categories, model-driven and data-driven (e.g., machine learning, ML), while each suffers from either deficient physics or small data. To mitigate these limitations, recent studies introduced a hybrid paradigm, physics-informed deep learning (PIDL), which contains both model-driven and data-driven components. This paper contributes an improved version, called physics-informed deep learning with a fundamental diagram learner (PIDL+FDL), which integrates ML terms into the model-driven component to learn a functional form of a fundamental diagram (FD), i.e., a mapping from traffic density to flow or velocity. The proposed PIDL+FDL has the advantages of performing the TSE learning, model parameter identification, and FD estimation simultaneously. We demonstrate the use of PIDL+FDL to solve popular first-order and second-order traffic flow models and reconstruct the FD relation as well as model parameters that are outside the FD terms. We then evaluate the PIDL+FDL-based TSE using the Next Generation SIMulation (NGSIM) dataset. The experimental results show the superiority of the PIDL+FDL in terms of improved estimation accuracy and data efficiency over advanced baseline TSE methods, and additionally, the capacity to properly learn the unknown underlying FD relation.

cs.LG↗

arXiv2021

Physics-Informed Deep Learning for Traffic State Estimation

Traffic state estimation (TSE), which reconstructs the traffic variables (e.g., density) on road segments using partially observed data, plays an important role on efficient traffic control and operation that intelligent transportation systems (ITS) need to provide to people. Over decades, TSE approaches bifurcate into two main categories, model-driven approaches and data-driven approaches. However, each of them has limitations: the former highly relies on existing physical traffic flow models, such as Lighthill-Whitham-Richards (LWR) models, which may only capture limited dynamics of real-world traffic, resulting in low-quality estimation, while the latter requires massive data in order to perform accurate and generalizable estimation. To mitigate the limitations, this paper introduces a physics-informed deep learning (PIDL) framework to efficiently conduct high-quality TSE with small amounts of observed data. PIDL contains both model-driven and data-driven components, making possible the integration of the strong points of both approaches while overcoming the shortcomings of either. This paper focuses on highway TSE with observed data from loop detectors, using traffic density as the traffic variables. We demonstrate the use of PIDL to solve (with data from loop detectors) two popular physical traffic flow models, i.e., Greenshields-based LWR and three-parameter-based LWR, and discover the model parameters. We then evaluate the PIDL-based highway TSE using the Next Generation SIMulation (NGSIM) dataset. The experimental results show the advantages of the PIDL-based approach in terms of estimation accuracy and data efficiency over advanced baseline TSE methods.

cs.LG↗

arXiv2020

Dynamic driving and routing games for autonomous vehicles on networks: A mean field game approach

This paper aims to answer the research question as to optimal design of decision-making processes for autonomous vehicles (AVs), including dynamical selection of driving velocity and route choices on a transportation network. Dynamic traffic assignment (DTA) has been widely used to model travelers's route choice or/and departure-time choice and predict dynamic traffic flow evolution in the short term. However, the existing DTA models do not explicitly describe one's selection of driving velocity on a road link. Driving velocity choice may not be crucial for modeling the movement of human drivers but it is a must-have control to maneuver AVs. In this paper, we aim to develop a game-theoretic model to solve for AVs's optimal driving strategies of velocity control in the interior of a road link and route choice at a junction node. To this end, we will first reinterpret the DTA problem as an N-car differential game and show that this game can be tackled with a general mean field game-theoretic framework. The developed mean field game is challenging to solve because of the forward and backward structure for velocity control and the complementarity conditions for route choice. An efficient algorithm is developed to address these challenges. The model and the algorithm are illustrated on the Braess network and the OW network with a single destination. On the Braess network, we first compare the LWR based DTA model with the proposed game and find that the driving and routing control navigates AVs with overall lower costs. We then compare the total travel cost without and with the middle link and find that the Braess paradox may still arise under certain conditions. We also test our proposed model and solution algorithm on the OW network.

math.OC↗

arXiv2020

Stability of a Nonlocal Traffic Flow Model for Connected Vehicles

The emerging connected and automated vehicle technologies allow vehicles to perceive and process traffic information in a wide spatial range. Modeling nonlocal interactions between connected vehicles and analyzing their impact on traffic flows become important research questions to traffic planners. This paper considers a particular nonlocal LWR model that has been studied in the literature. The model assumes that vehicle velocities are controlled by the traffic density distribution in a nonlocal spatial neighborhood. By conducting stability analysis of the model, we obtain that, under suitable assumptions on how the nonlocal information is utilized, the nonlocal traffic flow is stable around the uniform equilibrium flow and all traffic waves dissipate exponentially. Meanwhile, improper use of the nonlocal information in the vehicle velocity selection could result in persistent traffic waves. Such results can shed light to the future design of driving algorithms for connected and automated vehicles.

math.AP↗

arXiv2020

Spectroscopically Confirmed Lyman-Alpha Emitters from Redshift 5 to 7 Behind Ten Galaxy Cluster Lenses

We present 36 spectroscopically confirmed intrinsically UV-faint Ly$\alpha$ emitting galaxies from follow-up observations with Keck/DEIMOS of gravitationally lensed high-redshift candidates. Candidates were selected to be between $5\lesssim z \lesssim 7$ from photometric data using \textit{HST} and \textit{Spitzer} imaging surveys. We used photometric redshift information to perform an integrated photometric redshift probability cut $>1\%$ between $5 $25{\AA} at $3\sigma$ at the fiducial depth of our survey, and only those galaxies with EW(Ly$\alpha$)$>$25{\AA} as true LAEs, and finally, only objects with $m_{AB}<26.8$, we found the LAE fraction to be flat, or modestly increase from 0.26$\pm0.04$ to 0.30$\pm0.04$. These values relative to those for lower-redshift samples are consistent with a rising LAE fraction with redshift out to $z\sim6$, but at $z\sim6.5$ there is some tension between our results and results from surveys at intrinsically brighter luminosities. We conclude intrinsically fainter galaxies have Ly$\alpha$ emission, and there is a steep drop in the LAE fraction from our high-redshift sample at $z\sim6.5$ and from similar galaxies at $z\sim7.5$. This likely indicates we are witnessing the tail end of the epoch of reionization, as such a drop is not expected due to changes of intrinsic galaxy properties between these redshifts.

astro-ph.CO↗