Signed particles and neural networks, towards efficient simulations of quantum systems

Sellier, Jean Michel; Caron, Gaétan Marceau; Leygonie, Jacob

doi:10.1016/j.jcp.2019.02.036

Cited by 8 publications

(4 citation statements)

References 16 publications

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“…Data-driven physical prediction. Data-driven approaches have been widely applied in physical systems including fluid mechanics [6], wave physics [17], quantum physics [30], thermodynamics [16], and material science [33]. Among these different physical systems, data-driven fluid receives increasing attention.…”

Section: Related Workmentioning

confidence: 99%

Nonseparable Symplectic Neural Networks

Xiong

Tong

et al. 2020

Preprint

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Predicting the behaviors of Hamiltonian systems has been drawing increasing attention in scientific machine learning. However, the vast majority of the literature was focused on predicting separable Hamiltonian systems with their kinematic and potential energy terms being explicitly decoupled, while building data-driven paradigms to predict nonseparable Hamiltonian systems that are ubiquitous in fluid dynamics and quantum mechanics were rarely explored. The main computational challenge lies in the effective embedding of symplectic priors to describe the inherently coupled evolution of position and momentum, which typically exhibits intricate dynamics with many degrees of freedom. To solve the problem, we propose a novel neural network architecture, Nonseparable Symplectic Neural Networks (NSSNNs), to uncover and embed the symplectic structure of a nonseparable Hamiltonian system from limited observation data. The enabling mechanics of our approach is an augmented symplectic time integrator to decouple the position and momentum energy terms and facilitate their evolution. We demonstrated the efficacy and versatility of our method by predicting a wide range of Hamiltonian systems, both separable and nonseparable, including vortical flow and quantum system. We showed the unique computational merits of our approach to yield longterm, accurate, and robust predictions for large-scale Hamiltonian systems by rigorously enforcing symplectomorphism.

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Section: Related Workmentioning

confidence: 99%

Nonseparable Symplectic Neural Networks

Xiong

Tong

et al. 2020

Preprint

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“…The rapid advent of machine learning techniques is opening up new possibilities to solve the physical system's identification problems by statistically exploring the underlying structure of a variety of physical systems, encompassing applications in quantum physics [35], thermodynamics [14], material science [36], rigid body control [9], Lagrangian systems [8], and Hamiltonian systems [10,19,37]. Specifically, in the field of fluid mechanics, machine learning offers a wealth of techniques to extract information from data that could be translated into knowledge about the underlying fluid field, as well as exhibits its ability to augment domain knowledge and automate tasks related to flow control and optimization [4,11].…”

Section: Machine Learning In Fluid Systemsmentioning

confidence: 99%

Neural Vortex Method: from Finite Lagrangian Particles to Infinite Dimensional Eulerian Dynamics

Xiong¹,

He²,

Tong³

et al. 2020

Preprint

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In the field of fluid numerical analysis, there has been a long-standing problem: lacking of a rigorous mathematical tool to map from a continuous flow field to discrete vortex particles, hurdling the Lagrangian particles from inheriting the high resolution of a large-scale Eulerian solver. To tackle this challenge, we propose a novel learning-based framework, the Neural Vortex Method (NVM), which builds a neural-network description of the Lagrangian vortex structures and their interaction dynamics to reconstruct the high-resolution Eulerian flow field in a physically-precise manner. The key components of our infrastructure consist of two networks: a vortex representation network to identify the Lagrangian vortices from a grid-based velocity field and a vortex interaction network to learn the underlying governing dynamics of these finite structures. By embedding these two networks with a vorticity-to-velocity Poisson solver and training its parameters using the high-fidelity data obtained from high-resolution direct numerical simulation, we can predict the accurate fluid dynamics on a precision level that was infeasible for all the previous conventional vortex methods (CVMs). To the best of our knowledge, our method is the first approach that can utilize motions of finite particles to learn infinite dimensional dynamic systems. We demonstrate the efficacy of our method in generating highly accurate prediction results, with low computational cost, of the leapfrogging vortex rings system, the turbulence system, and the systems governed by Euler equations with different external forces.

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“…Due to its data-driven nature, a well-trained NNW model can learn to represent or encode each data point in a given large data set in terms of a more compact vector in internal (hidden) dimensions. DL thus can be efficiently applied for several types of tasks, such as approximating quantum wave functions [14][15][16][17][18][19][20][21][22][23] , assisting quantum simulations [24][25][26][27][28][29] , and detecting phases of matter [30][31][32][33][34][35][36][37][38][39][40][41] . In particular, DL has be shown not only to help recognize conventional, symmetry-breaking phase transitions but also to discover non-local, topological ones [42][43][44][45][46][47][48][49][50] .…”

Section: Introductionmentioning

confidence: 99%

Deep learning of topological phase transitions from entanglement aspects for two-dimensional chiral p-wave superconductors

Chung,

Cheng,

Huang

et al. 2021

Preprint

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Applying deep learning to investigate topological phase transitions (TPTs) becomes a useful method due to not only its ability to recognize patterns but also its statistical excellency to examine the mount of information carried by different types of data inputs. Among possible data types, entanglement-related quantities, such as Majorana correlation matrices (MCMs), one-particle entanglement spectra (OPES), and entanglement eigenvectors (OPEEs), have been proved effective, however, are to date mostly restricted to one dimension. Here, we propose practical input data forms based on those quantities to study TPTs and to compare the efficiency of each form on classic two-dimensional chiral p-wave superconductors via the deep learning approach. First, we find that different input forms, either matrices or tensors both originated from real MCMs, can affect the precise locations of the predicted transition points. Next, due to the complex nature of OPEEs, we extract three spatially dependent quantities from OPEEs, one related to the "intensity", and the other two related to "phases" of particle and hole components. We show that similar to taking OPES directly as inputs, solely using "intensity" quantity can only distinguish topological phases from trivial ones, whereas using either whole MCMs or complete OPEE-extracted quantities can provide sufficient information for deep learning to distinguish between phases of matter with different U (1) gauges or Chern numbers. Finally, we discuss certain characteristic features in the deep learning approach and, in particular, they reveal that our trained models indeed learn physically meaningful features, which confirms the potential use even at high dimensions.

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Signed particles and neural networks, towards efficient simulations of quantum systems

Cited by 8 publications

References 16 publications

Nonseparable Symplectic Neural Networks

Nonseparable Symplectic Neural Networks

Neural Vortex Method: from Finite Lagrangian Particles to Infinite Dimensional Eulerian Dynamics

Deep learning of topological phase transitions from entanglement aspects for two-dimensional chiral p-wave superconductors

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