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Long-Time Dynamics of Differential Equations in Physics and AI
Long-Time Dynamics of Differential Equations in Physics and AI
상세정보
- 자료유형
- 학위논문 서양
- 최종처리일시
- 20260202103003
- ISBN
- 9798280749238
- DDC
- 510
- 저자명
- Jin, Kexin.
- 서명/저자
- Long-Time Dynamics of Differential Equations in Physics and AI
- 발행사항
- [Sl] : Princeton University, 2025
- 발행사항
- Ann Arbor : ProQuest Dissertations & Theses, 2025
- 형태사항
- 251 p
- 주기사항
- Source: Dissertations Abstracts International, Volume: 86-12, Section: B.
- 주기사항
- Advisor: Ionescu, Alexandru.
- 학위논문주기
- Thesis (Ph.D.)--Princeton University, 2025.
- 초록/해제
- 요약This thesis consists of two parts: the study of long-time dynamics of differential equations derived from 1) physics, and 2) machine learning.In the first part, we study the cubic nonlinear Schrodinger (NLS) equation with periodic boundary conditions defined on [0, L]. By proving a vanishing property of the normal form transformation and applying it to the quintic resonance interactions, we obtain a description of the dynamics for a time up to T = L 2/ε4 , where ϵ is the size of the initial data. Since T is the characteristic time of wave turbulence, this result implies the absence of wave turbulence behavior of the 1D cubic NLS. In the proof, we develop a correspondence between Feynman diagrams and terms in normal forms, which allows us to calculate the coefficients inductively. Notably, our approach can be adapted to other integrable systems with minimal difficulty.In the second part, we consider differential equations in machine learning algorithms. We develop a novel approach to model discrete-time machine learning optimization algorithms as continuous-time dynamics, including stochastic gradient descent (SGD) and its variants. We propose the stochastic gradient process consists in a gradient flow minimizing an indexed target function that is coupled with a continuous-time index process determining the index. Index processes are, e.g., reflected diffusions, pure jump processes, or other Levy processes on compact spaces. We analyze the approximation properties of the stochastic gradient process and study its long-time behavior and ergodicity under constant and decreasing learning rates. We illustrate the applicability of the stochastic gradient process in a polynomial regression problem with noisy functional data, as well as in a physics-informed neural network.
- 일반주제명
- Mathematics
- 일반주제명
- Applied mathematics
- 일반주제명
- Computational physics
- 키워드
- Machine learning
- 키워드
- Feynman diagrams
- 키워드
- Ergodicity
- 기타저자
- Princeton University Mathematics
- 기본자료저록
- Dissertations Abstracts International. 86-12B.
- 전자적 위치 및 접속
- 로그인 후 원문을 볼 수 있습니다.
MARC
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■006m o d
■007cr#unu||||||||
■020 ▼a9798280749238
■035 ▼a(MiAaPQ)AAI31840491
■040 ▼aMiAaPQ▼cMiAaPQ
■0820 ▼a510
■1001 ▼aJin, Kexin.
■24510▼aLong-Time Dynamics of Differential Equations in Physics and AI
■260 ▼a[Sl]▼bPrinceton University▼c2025
■260 1▼aAnn Arbor▼bProQuest Dissertations & Theses▼c2025
■300 ▼a251 p
■500 ▼aSource: Dissertations Abstracts International, Volume: 86-12, Section: B.
■500 ▼aAdvisor: Ionescu, Alexandru.
■5021 ▼aThesis (Ph.D.)--Princeton University, 2025.
■520 ▼aThis thesis consists of two parts: the study of long-time dynamics of differential equations derived from 1) physics, and 2) machine learning.In the first part, we study the cubic nonlinear Schrodinger (NLS) equation with periodic boundary conditions defined on [0, L]. By proving a vanishing property of the normal form transformation and applying it to the quintic resonance interactions, we obtain a description of the dynamics for a time up to T = L 2/ε4 , where ϵ is the size of the initial data. Since T is the characteristic time of wave turbulence, this result implies the absence of wave turbulence behavior of the 1D cubic NLS. In the proof, we develop a correspondence between Feynman diagrams and terms in normal forms, which allows us to calculate the coefficients inductively. Notably, our approach can be adapted to other integrable systems with minimal difficulty.In the second part, we consider differential equations in machine learning algorithms. We develop a novel approach to model discrete-time machine learning optimization algorithms as continuous-time dynamics, including stochastic gradient descent (SGD) and its variants. We propose the stochastic gradient process consists in a gradient flow minimizing an indexed target function that is coupled with a continuous-time index process determining the index. Index processes are, e.g., reflected diffusions, pure jump processes, or other Levy processes on compact spaces. We analyze the approximation properties of the stochastic gradient process and study its long-time behavior and ergodicity under constant and decreasing learning rates. We illustrate the applicability of the stochastic gradient process in a polynomial regression problem with noisy functional data, as well as in a physics-informed neural network.
■590 ▼aSchool code: 0181.
■650 4▼aMathematics
■650 4▼aApplied mathematics
■650 4▼aComputational physics
■653 ▼aMachine learning
■653 ▼aNonlinear Schrodinger equation
■653 ▼aFeynman diagrams
■653 ▼aStochastic gradient descent
■653 ▼aErgodicity
■690 ▼a0405
■690 ▼a0800
■690 ▼a0216
■690 ▼a0364
■71020▼aPrinceton University▼bMathematics.
■7730 ▼tDissertations Abstracts International▼g86-12B.
■790 ▼a0181
■791 ▼aPh.D.
■792 ▼a2025
■793 ▼aEnglish
■85640▼uhttp://www.riss.kr/pdu/ddodLink.do?id=T17356615▼nKERIS▼z이 자료의 원문은 한국교육학술정보원에서 제공합니다.
