In this laboratory, we are many interested in theoretic and applied mathematics as well as multidisciplinary researches.
Research themes for undergraduate and postgraduate theses are available below.
Applied Information Geometry
Classical information geometry studies the geometry of statistical manifolds, each point of which corresponds to a probability distribution. Since probability theory and statistics are used in many problems arising from, for instance, data analysis, optimisation, machine learning and neural networks, information geometry has been widely applied in the last decades.
情報幾何学では、統計的多様体の幾何学を研究し、その各点は確率分布に対応しています。 確率論と統計は、データ分析、最適化、機械学習、ニューラルネットワークなどから生じる多くの問題で用いられているため、様々な分野に情報幾何学を応用することができます。
Research themes
- Applications of information geometry to machine learning and data analysis
- Matrix informations geometry
- The construction of efficient numerical methods on manifolds
Some recent publications:
- (1) X. Hua, L. Peng, W. Liu, Y. Cheng, H. Wang, H. Sun, and Z. Wang, LDA-MIG Detectors for Maritime Targets in Nonhomogeneous Sea Clutter, IEEE Transactions on Geoscience and Remote Sensing 61: 5101815, 2023. PDF
- (2) Y. Ono and L. Peng, Towards a median signal detector through the total Bregman divergence and robustness analysis, Signal Processing 201: 108728, 2022. PDF
- (3) X. Hua, Y. Ono, L. Peng, and Y. Xu, Unsupervised Learning Discriminative MIG Detectors in Nonhomogeneous Clutter, IEEE Transactions on Communications 70: 4107-4120, 2022. PDF
- (4) X. Hua, Y. Ono, L. Peng, Y. Cheng, and H. Wang, Target Detection within Nonhomogeneous Clutter via Total Bregman Divergence-Based Matrix Information Geometry Detectors, IEEE Transactions on Signal Processing 69: 4326-4340, 2021. PDF
- (5) X. Hua and L. Peng, MIG Median Detectors with Manifold Filter, Signal Processing 188: 108176, 2021. PDF
Geometric Numerical Integrator
Unfortunately, only the simplest differential equations are solvable by explicit formulae. In many situations, we must seek help from numerical techniques. Our laboratory is particularly interested in geometric integrator, i.e., numerical methods that preserve geometric properties of the differential equations. These geometric properties can be symmetries, conservation laws, symplectic structures, multisymplectic structures, Dirac structure, etc.
複雑な微分方程式の場合、明示公式だけではそれらを解くことができません。当研究室では、この問題に対して、Geometric Integratorと呼ばれる微分方程式の幾何学的特性、例えば対称性、保存則、シンプレクティック構造、マルチシンプレクティック構造、ディラック構などを保存するような数値解法を用いてアプローチしています。
Research themes
- Variational integrator for fluid mechanics
- Symplectic/multisymplectic integrator
- Symplectic neural networks
- Discrete moving frames
Some recent publications:
- (1) 大竹 桃子, 彭 林玉, 変分的積分法を用いたベイジアンニューラルネットワークの応用, 計算力学講演会講演論文集2022.35卷 (2022), 4pp. PDF
- (2) 吉村 浩明, 彭 林玉, ディラック力学系のモデリングと離散的変分法による定式化, 計算力学講演会講演論文集2022.35卷 (2022), 4pp. PDF
- (3) L. Peng, N. Arai, and K. Yasuoka, A stochastic Hamiltonian formulation applied to dissipative particle dynamics, Applied Mathematics and Computation 426: 127126, 2022. PDF
- (4) E.L. Mansfield, A. Rojo-Echeburua, P.E. Hydon, and L. Peng, Moving frames and Noether’s finite difference conservation laws I, Transactions of Mathematics and its Applications 3: tnz004, 47pp, 2019. PDF
- (5) J.A. Wright and L. Peng, An automatic dynamic balancer in a rotating mechanism with time-varying angular velocity, Results in Applied Mathematics 2: 100015, 13pp, 2019. PDF
Symmetries of Differential Equations and Discrete Equations
Physical phenomena can often be modeled as differential equations or discrete equations, and hence the study of such equations is fundamentally important for us to understand the real world. Symmetries are among the most important techniques for solving the equations or revealing their solvability/integrability. In simple words, a symmetry group of a system of differential/discrete equations is a group of continuous (local) transformations moving one of its solutions to another solution.
Many important physical and mechanical problems can be formulated as variational systems, that are governed by least actions, called variational problems. The celebrated Noether’s theorem connects symmetries of least actions with conservation laws of the equations of motion. For instance, time translation symmetry corresponds to the conservation of energy, rotational symmetry corresponds to the conservation of angular momentum, etc.
物理現象は微分方程式や離散方程式としてモデル化できることが多いため、これらの方程式について研究することで現実の様々な現象への理解を深めることができます。 対称性は、方程式を解いたり、それらの可積分性を明らかにしたりするための最も重要な手法の1つです。 簡単に言えば、微分/離散方程式系の対称性は、その解の1つを別の解に移動する連続（局所）変換のリー群で表されます。
多くの重要な物理的および工学的問題は変分系とみなすことができるため、作用汎関数の最小作用の原理を用いて定式化することができます。 また有名なネーター定理により、最小の作用汎関数の対称性から保存則を導くことができます。 たとえば、時間並進対称性はエネルギー保存則に対応し、回転対称性は角運動量の保存則に対応します。
Research themes
- Symmetry analysis of differential equations, finite difference equations and semi-discrete equations
- Symmetries and fluid mechanics
- Symmetries applied to information geometry
- Differential and difference invariants, moving frames
Some recent publications:
- (1) L. Peng, A modified formal Lagrangian formulation for general differential equations, Japan Journal of Industrial and Applied Mathematics 39: 573-598, 2022. PDF
- (2) L. Peng and P. E. Hydon, Transformations, symmetries and Noether theorems for differential-difference equations, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 478: 20210944, 2022.
PDF
- (3) L. Peng, Symmetries and reductions of integrable nonlocal partial differential equations, Symmetry 11: 884, 11pp, 2019. PDF
- (4) L. Peng and Z. Zhang, Statistical Einstein manifolds of exponential families with group-invariant potential functions, J. Math. Anal. Appl. 479: 2104-2118, 2019. PDF
- (5) L. Peng, Symmetries, conservation laws, and Noether's theorem for differential-difference equations, Stud. Appl. Math. 139: 457-502, 2017. PDF