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Lax pairs informed neural networks solving integrable systems

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arXiv 2024年

作者： Pu, Juncai Chen, Yong Institute of Applied Physics and Computational Mathematics Beijing100094 China School of Mathematical Sciences Key Laboratory of Mathematics and Engineering Applications Ministry of Education Shanghai Key Laboratory of PMMP East China Normal University Shanghai200241 China College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao266590 China

Lax pairs are one of the most important features of integrable system. In this work, we propose the Lax pairs informed neural networks (LPNNs) tailored for the integrable systems with Lax pairs by designing novel network architectures and loss functions, comprising LPNN-v1 and LPNN-v2. The most noteworthy advantage of LPNN-v1 is that it can transform the solving of nonlinear integrable systems into the solving of a linear Lax pairs spectral problems, and it not only efficiently solves data-driven localized wave solutions, but also obtains spectral parameter and corresponding spectral function in Lax pairs spectral problems of the integrable systems. On the basis of LPNN-v1, we additionally incorporate the compatibility condition/zero curvature equation of Lax pairs in LPNN-v2, its major advantage is the ability to solve and explore high-accuracy data-driven localized wave solutions and associated spectral problems for integrable systems with Lax pairs. The numerical experiments focus on studying abundant localized wave solutions for very important and representative integrable systems with Lax pairs spectral problems, including the soliton solution of the Korteweg-de Vries (KdV) euqation and modified KdV equation, rogue wave solution of the nonlinear Schrödinger equation, kink solution of the sine-Gordon equation, non-smooth peakon solution of the Camassa-Holm equation and pulse solution of the short pulse equation, as well as the line-soliton solution of Kadomtsev-Petviashvili equation and lump solution of high-dimensional KdV equation. The innovation of this work lies in the pioneering integration of Lax pairs informed of integrable systems into deep neural networks, thereby presenting a fresh methodology and pathway for investigating data-driven localized wave solutions and Lax pairs spectral problems. Copyright © 2024, The Authors. All rights reserved.

关键词： Korteweg-de Vries equation

A GNN-GUIDED PREDICT-AND-SEARCH FRAMEWORK FOR MIXED-INTEGER LINEAR PROGRAMMING

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arXiv 2023年

作者： Han, Qingyu Yang, Linxin Chen, Qian Zhou, Xiang Zhang, Dong Wang, Akang Sun, Ruoyu Luo, Xiaodong Shenzhen Research Institute of Big Data China Shandong University China School of Data Science The Chinese University of Hong Kong Shenzhen China School of Science and Engineering The Chinese University of Hong Kong Shenzhen China Huawei China Shenzhen International Center For Industrial and Applied Mathematics Shenzhen Research Institute of Big Data China

Mixed-integer linear programming (MILP) is widely employed for modeling combinatorial optimization problems. In practice, similar MILP instances with only coefficient variations are routinely solved, and machine learning (ML) algorithms are capable of capturing common patterns across these MILP instances. In this work, we combine ML with optimization and propose a novel predict-and-search framework for efficiently identifying high-quality feasible solutions. Specifically, we first utilize graph neural networks to predict the marginal probability of each variable, and then search for the best feasible solution within a properly defined ball around the predicted solution. We conduct extensive experiments on public datasets, and computational results demonstrate that our proposed framework achieves 51.1% and 9.9% performance improvements to MILP solvers SCIP and Gurobi on primal gaps, respectively. © 2023, CC BY-NC-SA.

关键词： Machine learning

Analog Photonics Computing for Information Processing, Inference and Optimisation

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arXiv 2023年

作者： Stroev, Nikita Berloff, Natalia G. Department of Physics of Complex Systems Weizmann Institute of Science Rehovot76100 Israel Department of Applied Mathematics and Theoretical Physics University of Cambridge CambridgeCB3 0WA United Kingdom

This review presents an overview of the current state-of-the-art in photonics computing, which leverages photons, photons coupled with matter, and optics-related technologies for effective and efficient computational purposes. It covers the history and development of photonics computing and modern analogue computing platforms and architectures, focusing on optimization tasks and neural network implementations. The authors examine special-purpose optimizers, mathematical descriptions of photonics optimizers, and their various interconnections. Disparate applications are discussed, including direct encoding, logistics, finance, phase retrieval, machine learning, neural networks, probabilistic graphical models, and image processing, among many others. The main directions of technological advancement and associated challenges in photonics computing are explored, along with an assessment of its efficiency. Finally, the paper discusses prospects and the field of optical quantum computing, providing insights into the potential applications of this technology. © 2023, CC BY.

关键词： Optical data processing

Rényi entanglement entropy of a spin chain with generative neural networks

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Physical Review E 2024年第4期110卷 044116页

作者： Piotr Białas Piotr Korcyl Tomasz Stebel Dawid Zapolski Institute of Applied Computer Science <a href="https://***/03bqmcz70">Jagiellonian University</a> ul. Łojasiewicza 11 30-348 Kraków Poland. Institute of Theoretical Physics <a href="https://***/03bqmcz70">Jagiellonian University</a> ul. Łojasiewicza 11 30-348 Kraków Poland. Faculty of Mathematics and Computer Science <a href="https://***/03bqmcz70">Jagiellonian University</a> ul. Łojasiewicza 6 30-348 Kraków Poland.

We describe a method to estimate Rényi entanglement entropy of a spin system which is based on the replica trick and generative neural networks with explicit probability estimation. It can be extended to any spin system or lattice field theory. We demonstrate our method on a one-dimensional quantum Ising spin chain. As the generative model, we use a hierarchy of autoregressive networks, allowing us to simulate up to 32 spins. We calculate the second Rényi entropy and its derivative and cross-check our results with the numerical evaluation of entropy and results available in the literature.

关键词： Entanglement entropy Quantum statistical mechanics Deep learning Information theory Ising model Monte Carlo methods Neural network simulations

Positivity and Maximum Principle Preserving Discontinuous Galerkin Finite Element Schemes for a Coupled Flow and Transport

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arXiv 2024年

作者： Gong, Shihua Lee, Young-Ju Li, Yukun Yu, Yue School of Science and Engineering The Chinese University of Hong Kong Guangdong Shenzhen518172 China Department of Mathematics Texas State University TX78666 United States Department of Mathematics University of Central Florida United States Shenzhen International Center For Industrial And Applied Mathematics Shenzhen Research Institute of Big Data China

We introduce a new concept of the locally conservative flux and investigate its relationship with the compatible discretization pioneered by Dawson, Sun and Wheeler [11]. We then demonstrate how the new concept of the locally conservative flux can play a crucial role in obtaining the L2 norm stability of the discontinuous Galerkin finite element scheme for the transport in the coupled system with flow. In particular, the lowest order discontinuous Galerkin finite element for the transport is shown to inherit the positivity and maximum principle when the locally conservative flux is used, which has been elusive for many years in literature. The theoretical results established in this paper are based on the equivalence between Lesaint-Raviart discontinuous Galerkin scheme and Brezzi-Marini-Süli discontinuous Galerkin scheme for the linear hyperbolic system as well as the relationship between the Lesaint-Raviart discontinuous Galerkin scheme and the characteristic method along the streamline. Sample numerical experiments have also been performed to justify our theoretical findings. © 2024, CC BY.

关键词： Maximum principle

Iterative Thresholding Methods for Longest Minimal Length Partitions

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arXiv 2024年

作者： Hu, Shilong Liu, Hao Wang, Dong School of Science and Engineering The Chinese University of Hong Kong Guangdong Shenzhen518172 China Department of Mathematics Hong Kong Baptist University Kowloon Tong Hong Kong Shenzhen International Center for Industrial and Applied Mathematics Shenzhen Research Institute of Big Data Guangdong 518172 China

In this paper, we introduce two iterative methods for longest minimal length partition problem, which asks whether the disc (ball) is the set maximizing the total perimeter of the shortest partition that divides the total region into sub-regions with given volume proportions, under a volume constraint. The objective functional is approximated by a short-time heat flow using indicator functions of regions and Gaussian convolution. The problem is then represented as a constrained max-min optimization problem. Auction dynamics is used to find the shortest partition in a fixed region, and threshold dynamics is used to update the region. Numerical experiments in two-dimensional and three-dimensional cases are shown with different numbers of partitions, unequal volume proportions, and different initial shapes. The results of both methods are consistent with the conjecture that the disc in two dimensions and the ball in three dimensions are the solution of the longest minimal length partition *** Codes 49M20, 49Q05, 49M05, 41A30 Copyright © 2024, The Authors. All rights reserved.

关键词： Iterative methods

Robust Network Optimization by Deep Generative Models and Stochastic Optimization

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IEEE Transactions on Wireless Communications 2025年

作者： Zhang, Shutao Xue, Ye Tang, Zhiwei Wang, Hao Shen, Chao Shi, Qingjiang Chang, Tsung-Hui The Chinese University of Hon g Kong School of Science and Engineering Shenzhen China Shenzhen Research Institute of Big Data Shenzhen China The Chinese University of Hon g Kong School of Data Science Shenzhen China Shenzhen Research Institute of Big Data Shenzhen International Center for Industrial and Applied Mathematics China Guangdong Provincial Engineering Technology Research Center for Intelligent Industrial Digital Twin and Optimization Shenzhen518172 China Tongji University School of Software Engineering Shanghai China

Wireless network optimization is essential for improving the network performance in mobile communications. However, due to the stochastic nature of wireless networks, existing schemes based on analytical models and deterministic optimization are less reliable. To this end, we design a framework for robust network optimization based on deep generative models and stochastic optimization. Inspired by the powerful diffusion process, we propose a deep generative simulator to capture the statistical distribution of the network performance. By sampling from the deep generative simulator, we can alleviate the inherent uncertainty related to the network performance and devise an innovative expectation-quantile-based stochastic objective function. The inner expectation is designed for the temporal statistics, while the outer quantile is developed for the spatial statistics. This designated two-tier objective function is capable of mitigating temporal fluctuations and ensuring satisfactory network performance across most geographical grids, thereby achieving robustness. To solve this stochastic optimization problem, a smooth zeroth-order approach is introduced by taking advantage of the unique structure of quantile functions. Through theoretical performance analysis and simulation experiments with real-world datasets, we demonstrate the superiority of our approach over other baseline schemes, highlighting its practical utility in robust network optimization. © 2002-2012 IEEE.

关键词： Generative diffusion model quantile function robust approach wireless network optimization

CONVERGENCE ANALYSIS OF THE TRANSFORMED GRADIENT PROJECTION ALGORITHMS ON COMPACT MATRIX MANIFOLDS

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arXiv 2024年

作者： Ding, Wentao Li, Jianze Zhang, Shuzhong School of Data Science Shenzhen Research Institute of Big Data The Chinese University of Hong Kong Guangdong Shenzhen China Shenzhen International Center for Industrial and Applied Mathematics Shenzhen Research Institute of Big Data The Chinese University of Hong Kong Guangdong Shenzhen China Department of Industrial and Systems Engineering University of Minnesota MinneapolisMN55455 United States

In this paper, to address the optimization problem on a compact matrix manifold, we introduce a novel algorithmic framework called the Transformed Gradient Projection (TGP) algorithm, using the projection onto this compact matrix manifold. Compared with the existing algorithms, the key innovation in our approach lies in the utilization of a new class of search directions and various stepsizes, including the Armijo, nonmonotone Armijo, and fixed stepsizes, to guide the selection of the next iterate. Our framework offers flexibility by encompassing the classical gradient projection algorithms as special cases, and intersecting the retraction-based line-search algorithms. Notably, our focus is on the Stiefel or Grassmann manifold, revealing that many existing algorithms in the literature can be seen as specific instances within our proposed framework, and this algorithmic framework also induces several new special cases. Then, we conduct a thorough exploration of the convergence properties of these algorithms, considering various search directions and stepsizes. To achieve this, we extensively analyze the geometric properties of the projection onto compact matrix manifolds, allowing us to extend classical inequalities related to retractions from the literature. Building upon these insights, we establish the weak convergence, convergence rate, and global convergence of TGP algorithms under three distinct stepsizes. In cases where the compact matrix manifold is the Stiefel or Grassmann manifold, our convergence results either encompass or surpass those found in the literature. Finally, through a series of numerical experiments, we observe that the TGP algorithms, owing to their increased flexibility in choosing search directions, outperform classical gradient projection and retraction-based line-search algorithms in several *** Codes 15A23, 49M37, 65K05, 90C26, 90C30 Copyright © 2024, The Authors. All rights reserved.

关键词： Optimization

A Fast and Accurate Solver for the Fractional Fokker-Planck Equation with Dirac-Delta Initial Conditions

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arXiv 2024年

作者： Ye, Qihao Tian, Xiaochuan Wang, Dong Department of Mathematics University of California San DiegoCA92093 United States School of Science and Engineering The Chinese University of Hong Kong Shenzhen Guangdong518172 China Shenzhen International Center for Industrial and Applied Mathematics Shenzhen Research Institute of Big Data Guangdong518172 China

The classical Fokker-Planck equation (FPE) is a key tool in physics for describing systems influenced by drag forces and Gaussian noise, with applications spanning multiple fields. We consider the fractional Fokker-Planck equation (FFPE), which models the time evolution of probability densities for systems driven by Lévy processes, relevant in scenarios where Gaussian assumptions fail. The paper presents an efficient and accurate numerical approach for the free-space FFPE with constant coefficients and Dirac-delta initial conditions. This method utilizes the integral representation of the solutions and enables the efficient handling of very high-dimensional problems using fast algorithms. Our work is the first to present a high-precision numerical solver for the free-space FFPE with Dirac-delta initial conditions. This opens the door for future research on more complex scenarios, including those with variable coefficients and other types of initial *** Codes 34K37, 44A35, 35Q84, 65D40, 33C10 Copyright © 2024, The Authors. All rights reserved.

关键词： Fokker Planck equation