tgEDMD: Approximation of the Kolmogorov Operator in Tensor Train Format

Marvin, Lücke,; Nüske, Feliks

doi:10.1007/s00332-022-09801-0

Cited by 5 publications

(2 citation statements)

References 47 publications

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“…A tensor-based formulation of gEDMD, which can be regarded as a combination of the methods presented in Klus et al (2020b) and Nüske et al (2021), for the approximation of the Koopman generator is described in Lücke and Nüske (2022).…”

Section: Tensor-based Variantsmentioning

confidence: 99%

Overcoming the timescale barrier in molecular dynamics: Transfer operators, variational principles and machine learning

Schütte¹,

Klus²,

Hartmann³

2023

Acta Numerica

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One of the main challenges in molecular dynamics is overcoming the ‘timescale barrier’: in many realistic molecular systems, biologically important rare transitions occur on timescales that are not accessible to direct numerical simulation, even on the largest or specifically dedicated supercomputers. This article discusses how to circumvent the timescale barrier by a collection of transfer operator-based techniques that have emerged from dynamical systems theory, numerical mathematics and machine learning over the last two decades. We will focus on how transfer operators can be used to approximate the dynamical behaviour on long timescales, review the introduction of this approach into molecular dynamics, and outline the respective theory, as well as the algorithmic development, from the early numerics-based methods, via variational reformulations, to modern data-based techniques utilizing and improving concepts from machine learning. Furthermore, its relation to rare event simulation techniques will be explained, revealing a broad equivalence of variational principles for long-time quantities in molecular dynamics. The article will mainly take a mathematical perspective and will leave the application to real-world molecular systems to the more than 1000 research articles already written on this subject.

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Section: Tensor-based Variantsmentioning

confidence: 99%

Overcoming the timescale barrier in molecular dynamics: Transfer operators, variational principles and machine learning

Schütte¹,

Klus²,

Hartmann³

2023

Acta Numerica

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“…[19][20][21] The idea behind this format is to decompose a high-dimensional tensor into a chain-like network of lower-dimensional tensors, which enables us to simulate and analyze large-scale problems if the underlying coupling structure allows for low-rank decompositions. Several applications of tensor trains-which can be considered as a special case of the ansatz used in the multi-layer (ML) variant of MCTDH 15,16 mentioned above-and tensor-train operators have shown that it is possible to mitigate the curse of dimensionality and to tackle high-dimensional problems, which cannot be solved using conventional numerical methods, see, e.g., dynamical systems, 22,23 system identification, 24,25 quantum mechanics, [26][27][28] and also quantum machine learning. 29,30 Typically, the applications require the approximation of the solutions of systems of linear equations, eigenvalue problems, and ordinary/partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

WaveTrain: A Python package for numerical quantum mechanics of chain-like systems based on tensor trains

Riedel

Gelß

Klein

et al. 2023

The Journal of Chemical Physics

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WaveTrain is an open-source software for numerical simulations of chain-like quantum systems with nearest-neighbor (NN) interactions only. The Python package is centered around tensor train (TT, or matrix product) format representations of Hamiltonian operators and (stationary or time-evolving) state vectors. It builds on the Python tensor train toolbox Scikit_tt, which provides efficient construction methods and storage schemes for the TT format. Its solvers for eigenvalue problems and linear differential equations are used in WaveTrain for the time-independent and time-dependent Schrödinger equations, respectively. Employing efficient decompositions to construct low-rank representations, the tensor-train ranks of state vectors are often found to depend only marginally on the chain length N. This results in the computational effort growing only slightly more than linearly with N, thus mitigating the curse of dimensionality. As a complement to the classes for full quantum mechanics, WaveTrain also contains classes for fully classical and mixed quantum–classical (Ehrenfest or mean field) dynamics of bipartite systems. The graphical capabilities allow visualization of quantum dynamics “on the fly,” with a choice of several different representations based on reduced density matrices. Even though developed for treating quasi-one-dimensional excitonic energy transport in molecular solids or conjugated organic polymers, including coupling to phonons, WaveTrain can be used for any kind of chain-like quantum systems, with or without periodic boundary conditions and with NN interactions only. The present work describes version 1.0 of our WaveTrain software, based on version 1.2 of scikit_tt, both of which are freely available from the GitHub platform where they will also be further developed. Moreover, WaveTrain is mirrored at SourceForge, within the framework of the WavePacket project for numerical quantum dynamics. Worked-out demonstration examples with complete input and output, including animated graphics, are available.

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Tensor-SqRA: Modeling the transition rates of interacting molecular systems in terms of potential energies

Sikorski,

Niknejad,

Weber

et al. 2024

The Journal of Chemical Physics

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Estimating the rate of rare conformational changes in molecular systems is one of the goals of molecular dynamics simulations. In the past few decades, a lot of progress has been done in data-based approaches toward this problem. In contrast, model-based methods, such as the Square Root Approximation (SqRA), directly derive these quantities from the potential energy functions. In this article, we demonstrate how the SqRA formalism naturally blends with the tensor structure obtained by coupling multiple systems, resulting in the tensor-based Square Root Approximation (tSqRA). It enables efficient treatment of high-dimensional systems using the SqRA and provides an algebraic expression of the impact of coupling energies between molecular subsystems. Based on the tSqRA, we also develop the projected rate estimation, a hybrid data-model-based algorithm that efficiently estimates the slowest rates for coupled systems. In addition, we investigate the possibility of integrating low-rank approximations within this framework to maximize the potential of the tSqRA.

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tgEDMD: Approximation of the Kolmogorov Operator in Tensor Train Format

Cited by 5 publications

References 47 publications

Overcoming the timescale barrier in molecular dynamics: Transfer operators, variational principles and machine learning

Overcoming the timescale barrier in molecular dynamics: Transfer operators, variational principles and machine learning

WaveTrain: A Python package for numerical quantum mechanics of chain-like systems based on tensor trains

Tensor-SqRA: Modeling the transition rates of interacting molecular systems in terms of potential energies

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