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Papers - PENG Linyu

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Stochastic multisymplectic PDEs and their structure-preserving numerical methods

Ruiao Hu, Linyu Peng

Studies in Applied Mathematics 155 ( 3 ) 2025.09

ISSN 00222526

We construct stochastic multisymplectic systems by considering a stochastic
extension to the variational formulation of multisymplectic partial
differential equations proposed in [Hydon, {\it Proc. R. Soc. A}, 461,
1627--1637, 2005]. The stochastic variational principle implies the existence
of stochastic $1$-form and $2$-form conservation laws, as well as conservation
laws arising from continuous variational symmetries via a stochastic Noether's
theorem. These results are the stochastic analogues of those found in
deterministic variational principles. Furthermore, we develop stochastic
structure-preserving collocation methods for this class of stochastic
multisymplectic systems. These integrators possess a discrete analogue of the
stochastic $2$-form conservation law and, in the case of linear systems, also
guarantee discrete momentum conservation. The effectiveness of the proposed
methods is demonstrated through their application to stochastic nonlinear
Schr\"odinger equations featuring either stochastic transport or stochastic
dispersion.

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Operator-based graph linear canonical transform

Jian-Yi Chen, Bing-Zhao Li, Linyu Peng

Journal of the Franklin Institute 362 ( 12 ) 2025.08

ISSN 00160032

This paper proposes a novel graph linear canonical transform framework based on hyper-differential operators, referred to as OGLCT. First, the relationship between the graph fractional Fourier transform (GFrFT) and the graph Fourier transform (GFT) is analyzed, and hyper-differential operators associated with the GFT matrix are derived through solutions to the Sylvester equation. Using these operators, the GFrFT, graph scaling, and graph chirp modulation are defined, culminating in the formal definition of the OGLCT and its fundamental properties. Next, the implementation of the OGLCT is discussed, and parameter optimization techniques are explored. Finally, the application of the OGLCT for feature extraction in image processing is explored, and its effectiveness in image classification tasks is investigated. By adjusting the parameters of the OGLCT, it is demonstrated that flexible tuning significantly enhances model performance, enabling more effective and versatile applications.

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The Effect of Accuracy of Initial Velocity Discretisations on Discrete Energy in Variational Integration

M Gunji, Y Ono, L Peng

Proceedings of the IUTAM Symposium on Nonlinear Dynamics for Design of … (Springer Cham) 88 - 98 2025.01

Euler--Poincar\'e reduction and the Kelvin--Noether theorem for discrete mechanical systems with advected parameters and additional dynamics

Y Ono, S Fiori, L Peng

arXiv preprint arXiv:2501.12940 2025

Reconstruction of Graph Signals on Complex Manifolds with Kernel Methods

Y Zhang, L Peng, BZ Li

arXiv preprint arXiv:2505.15202 2025

Modulated Diffusion: Accelerating Generative Modeling with Modulated Quantization

Weizhi Gao, Zhichao Hou, Junqi Yin, Feiyi Wang, Linyu Peng, Xiaorui Liu

arXiv preprint arXiv:2506.22463 (arXiv) 267 18337 - 18362 2025

Diffusion models have emerged as powerful generative models, but their high
computation cost in iterative sampling remains a significant bottleneck. In
this work, we present an in-depth and insightful study of state-of-the-art
acceleration techniques for diffusion models, including caching and
quantization, revealing their limitations in computation error and generation
quality. To break these limits, this work introduces Modulated Diffusion
(MoDiff), an innovative, rigorous, and principled framework that accelerates
generative modeling through modulated quantization and error compensation.
MoDiff not only inherents the advantages of existing caching and quantization
methods but also serves as a general framework to accelerate all diffusion
models. The advantages of MoDiff are supported by solid theoretical insight and
analysis. In addition, extensive experiments on CIFAR-10 and LSUN demonstrate
that MoDiff significant reduces activation quantization from 8 bits to 3 bits
without performance degradation in post-training quantization (PTQ). Our code
implementation is available at https://github.com/WeizhiGao/MoDiff.

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Neural bidomain model for multidimensional ephaptic coupling of neural spike propagation along myelinated fiber bundles with the nodes of Ranvier

S Chun, L Peng, HJ Park

bioRxiv 2024.12

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半離散方程式の対称性と群不変解—Symmetries and Group-Invariant Solutions of Semi-Discrete Equations—小特集エンジニアリングにおける応用数理の最前線 : 数理と工学の融合と信頼性を目指して

彭林玉, 富田繁, 郡司士

シミュレーション = Journal of the Japan Society for Simulation Technology / 日本シミュレーション学会編 (東京 : 小宮山印刷工業) 43 ( 4 ) 193 - 202 2024.12

ISSN 0285-9947

CiNii Research

Discrete Dirac structures and discrete Lagrange--Dirac dynamical systems in mechanics

Linyu Peng, Hiroaki Yoshimura

arXiv preprint arXiv:2411.09530 2024.11

In this paper, we propose the concept of $(\pm)$-discrete Dirac structures
over a manifold, where we define $(\pm)$-discrete two-forms on the manifold and
incorporate discrete constraints using $(\pm)$-finite difference maps.
Specifically, we develop $(\pm)$-discrete induced Dirac structures as discrete
analogues of the induced Dirac structure on the cotangent bundle over a
configuration manifold, as described by Yoshimura and Marsden (2006). We
demonstrate that $(\pm)$-discrete Lagrange--Dirac systems can be naturally
formulated in conjunction with the $(\pm)$-induced Dirac structure on the
cotangent bundle. Furthermore, we show that the resulting equations of motion
are equivalent to the $(\pm)$-discrete Lagrange--d'Alembert equations proposed
in Cort\'es and Mart\'inez (2001) and McLachlan and Perlmutter (2006). We also
clarify the variational structures of the discrete Lagrange--Dirac dynamical
systems within the framework of the $(\pm)$-discrete
Lagrange--d'Alembert--Pontryagin principle. Finally, we validate the proposed
discrete Lagrange--Dirac systems with some illustrative examples of
nonholonomic systems through numerical tests.

Information geometry and alpha-parallel prior of the beta-logistic distribution

L Jiu, L Peng

Communications in Statistics-Theory and Methods 2024.08

The Comparison of Riemannian Geometric Matrix-CFAR Signal Detectors

Y Ono, L Peng

IEEE Transactions on Aerospace and Electronic Systems 60 ( 2 ) 1679 - 1691 2024.04

ISSN 00189251

Essential characteristics of signal data can be captured by the autocovariance matrix, which, in the stationary scenarios, is Toeplitz Hermitian positive definite (HPD). In this paper, several well-known Riemannian geometric structures of HPD matrix manifolds are applied to signal detection, including the affine invariant Riemannian metric, the log-Euclidean metric, and the Bures–Wasserstein (BW) metric, the last of which was recently extended to HPD manifolds. Riemannian gradient descent algorithms are proposed to solve the corresponding geometric means and medians, that play fundamental roles in the detection process. Simulations within the scenario using the ideal steering vector as the target signal provide compelling evidence that the BW detectors outperform the other geometric detectors as well as the conventional adaptive matched filter and adaptive normalized matched filter when observation data are limited. Further simulations demonstrate that the matrix-CFAR is robust in scenarios where the signal is mismatched. In addition to detection performances, robustness of the geometric detectors to outliers and computational complexity of the algorithms are analysed.

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Developing a cloud evidence method for dynamic early warning of tunnel construction safety risk in undersea environment

H Zhou, B Gao, X Zhao, L Peng, S Bai

Developments in the Built Environment 16 100225 2023.12

ISSN 2666-1659

Traditional methods have limitations in achieving precise predictions of risk occurrence at an exact future time and have difficulties transforming between qualitative and quantitative indicators and handling multi-source heterogeneous risk data. This study quantifies and analyzes the multi-source construction safety risks classified into the categories of man, machine, material, method and environment (4M1E), and presents a cloud evidence method that integrates wavelet de-noising algorithm, cloud model, and Dempster-Shafer (D-S) evidence theory. A real-time risk prediction and warning is provided using this method after the fusion of multi-source uncertain information and the transformation between qualitative and quantitative indicators, enabling the timely detection of potential risks for project managers. This method analyzing “uncertainty” with “certainty” is verified by an undersea tunnel construction project. The result shows that this method is effective in early warning risks two days before their actual occurrence, providing reference significance for risk early warning of the tunnel construction project.

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The difference variational bicomplex and multisymplectic systems

Linyu Peng, Peter E. Hydon

arXiv preprint arXiv:2307.13935 2023.07

The difference variational bicomplex, which is the natural setting for
systems of difference equations, is constructed and used to examine the
geometric and algebraic properties of various systems. Exactness of the
bicomplex gives a coordinate-free setting for finite difference variational
problems, Euler--Lagrange equations and Noether's theorem. We also examine the
connection between the condition for existence of a Hamiltonian and the
multisymplecticity of systems of partial difference equations. Furthermore, we
define difference multimomentum maps of multisymplectic systems, which yield
their conservation laws. To conclude, we demonstrate how multisymplectic
integrators can be comprehended even on non-uniform meshes through a
generalized difference variational bicomplex.

Some novel physical structures of a (2+1)-dimensional variable-coefficient Korteweg–de Vries system

Yaqing Liu, Linyu Peng

Chaos, Solitons and Fractals 171 113430 2023.06

Corresponding author, Accepted, ISSN 09600779

In this paper, we study the novel nonlinear wave structures of a (2+1)-dimensional variable-coefficient Korteweg–de Vries (KdV) system by its analytic solutions. Its N-soliton solutions are obtained via Hirota's bilinear method, and in particular, the hybrid solutions of lump, breather and line solitons are derived by the long wave limit method. In addition to soliton solutions, similarity reduction, including similarity solutions (also known as group-invariant solutions) and novel non-autonomous rational third-order Painlevé equations, is achieved through symmetry analysis. The analytic results, together with illustrative wave interactions, show interesting physical features, that may shed some light on the study of other variable-coefficient nonlinear systems.

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LDA-MIG Detectors for Maritime Targets in Nonhomogeneous Sea Clutter

X Hua, L Peng, W Liu, Y Cheng, H Wang, H Sun, Z Wang

IEEE Transactions on Geoscience and Remote Sensing 61 5101815 2023.03

Accepted, ISSN 01962892

This article deals with the problem of detecting maritime targets embedded in nonhomogeneous sea clutter, where the limited number of secondary data is available due to the heterogeneity of sea clutter. A class of linear discriminant analysis (LDA)-based matrix information geometry (MIG) detectors is proposed in the supervised scenario. As customary, Hermitian positive-definite (HPD) matrices are used to model the observational sample data, and the clutter covariance matrix of the received dataset is estimated as the geometric mean of the secondary HPD matrices. Given a set of training HPD matrices with class labels, which are elements of a higher dimensional HPD matrix manifold, the LDA manifold projection learns a mapping from the higher dimensional HPD matrix manifold to a lower dimensional one subject to maximum discrimination. In this study, the LDA manifold projection, with the cost function maximizing between-class distance while minimizing within-class distance, is formulated as an optimization problem in the Stiefel manifold. Four robust LDA-MIG detectors corresponding to different geometric measures are proposed. Numerical results based on both simulated radar clutter with interferences and real IPIX radar data show the advantage of the proposed LDA-MIG detectors against their counterparts without using LDA and the state-of-the-art maritime target detection methods in nonhomogeneous sea clutter.

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The target detection method through autocovariance matrices and its robust analysis

Y Ono, L Peng

IEICE Technical Report; IEICE Tech. Rep. 122 ( 388 ) 55 - 60 2023.02

自己共分散行列空間における信号検出手法とそのロバスト性解析

小野悠介，彭林玉

研究報告音声言語情報処理 (SLP) 2023 (14), 1-6 2023-SLP-146 ( 14 ) 1 - 6 2023.02

Lagrangian Multiform Theory and Pluri-Lagrangian Systems (23w5043)

F Nijhoff, L Peng, Y Shi, D Zhang

2023

Towards a simple mathematical model for the legal concept of balancing of interests

Frederike Zufall, Rampei Kimura, Linyu Peng

Artificial Intelligence and Law 31 ( 4 ) 807 - 827 2023

Research paper (scientific journal), Accepted, ISSN 0924-8463

We propose simple nonlinear mathematical models for the legal concept of balancing of interests. Our aim is to bridge the gap between an abstract formalisation of a balancing decision while assuring consistency and ultimately legal certainty across cases. We focus on the conflict between the rights to privacy and to the protection of personal data in Art. 7 and Art. 8 of the EU Charter of Fundamental Rights (EUCh) against the right of access to information derived from Art. 11 EUCh. These competing rights are denoted by (i1) right to privacy and (i2) access to information; mathematically, their indices are respectively assigned by u1∈ [0 , 1] and u2∈ [0 , 1] subject to the constraint u1+ u2= 1. This constraint allows us to use one single index u to resolve the conflict through balancing. The outcome will be concluded by comparing the index u with a prior given threshold u. For simplicity, we assume that the balancing depends on only selected legal criteria such as the social status of affected person, and the sphere from which the information originated, which are represented as inputs of the models, called legal parameters. Additionally, we take “time” into consideration as a legal criterion, building on the European Court of Justice’s ruling on the right to be forgotten: by considering time as a legal parameter, we model how the outcome of the balancing changes over the passage of time. To catch the dependence of the outcome u by these criteria as legal parameters, data were created by a fully-qualified lawyer. By comparison to other approaches based on machine learning, especially neural networks, this approach requires significantly less data. This might come at the price of higher abstraction and simplification, but also provides for higher transparency and explainability. Two mathematical models for u, a time-independent model and a time-dependent model, are proposed, that are fitted by using the data.

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Towards a median signal detector through the total Bregman divergence and its robustness analysis

Y Ono, L Peng

Signal Processing 201 108728 2022.12

Accepted, ISSN 01651684

A novel family of geometric signal detectors are proposed through medians of the total Bregman divergence (TBD), which are shown advantageous over the conventional methods and their mean counterparts. By interpreting the observation data as Hermitian positive-definite matrices, their mean or median play an essential role in signal detection. As is difficult to be solved analytically, we propose numerical solutions through Riemannian gradient descent algorithms or fixed-point algorithms. Beside detection performance, robustness of a detector to outliers is also of vital importance, which can often be analyzed via the influence functions. Introducing an orthogonal basis for Hermitian matrices, we are able to compute the corresponding influence functions analytically and exactly by solving a linear system, which is transformed from the governing matrix equation. Numerical simulations show that the TBD medians are more robust than their mean counterparts.

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