| 1. |
- Bayer, C., et al.
(författare)
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Computational error estimates for Born-Oppenheimer molecular dynamics with nearly crossing potential surfaces
- 2015
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Ingår i: Applied Mathematics Research eXpress. - : Oxford University Press. - 1687-1200 .- 1687-1197. ; :2, s. 329-417
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Tidskriftsartikel (refereegranskat)abstract
- The difference of the values of observables for the time-independent Schrödinger equation, with matrix-valued potentials, and the values of observables for ab initio Born-Oppenheimer molecular dynamics, of the ground state, depends on the probability to be in excited states, and the electron/nuclei mass ratio. The paper first proves an error estimate (depending on the electron/nuclei mass ratio and the probability to be in excited states) for this difference of microcanonical observables, assuming that molecular dynamics space-time averages converge, with a rate related to the maximal Lyapunov exponent. The error estimate is uniform in the number of particles and the analysis does not assume a uniform lower bound on the spectral gap of the electron operator and consequently the probability to be in excited states can be large. A numerical method to determine the probability to be in excited states is then presented, based on Ehrenfest molecular dynamics, and stability analysis of a perturbed eigenvalue problem.
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| 2. |
- Bayer, Christian, et al.
(författare)
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How accurate is molecular dynamics?
- 2012
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- Born-Oppenheimer dynamics is shown to provide an accurate approximation of time-independent Schrödinger observables for a molecular system with an electron spectral gap, in the limit of large ratio of nuclei and electron masses, without assuming that the nuclei are localized to vanishing domains. The derivation, based on a Hamiltonian system interpretation of the Schrödinger equation and stability of the corresponding hitting time Hamilton-Jacobi equation for non ergodic dynamics, bypasses the usual separation of nuclei and electron wave functions, includes caustic states and gives a different perspective on theBorn-Oppenheimer approximation, Schrödinger Hamiltonian systems and numerical simulation in molecular dynamics modeling at constant energy.
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| 3. |
- Huang, Xin, et al.
(författare)
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Canonical mean-field molecular dynamics derived from quantum mechanics
- 2022
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Ingår i: ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS. - : EDP Sciences. - 2822-7840 .- 2804-7214. ; 56:6, s. 2197-2238
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Tidskriftsartikel (refereegranskat)abstract
- Canonical quantum correlation observables can be approximated by classical molecular dynamics. In the case of low temperature the ab initio molecular dynamics potential energy is based on the ground state electron eigenvalue problem and the accuracy has been proven to be O(M-1), provided the first electron eigenvalue gap is sufficiently large compared to the given temperature and M is the ratio of nuclei and electron masses. For higher temperature eigenvalues corresponding to excited electron states are required to obtain O(M-1) accuracy and the derivations assume that all electron eigenvalues are separated, which for instance excludes conical intersections. This work studies a mean-field molecular dynamics approximation where the mean-field Hamiltonian for the nuclei is the partial trace h := Tr(He-beta H)/Tr(e(-beta H)) with respect to the electron degrees of freedom and H is the Weyl symbol corresponding to a quantum many body Hamiltonian (sic). It is proved that the mean-field molecular dynamics approximates canonical quantum correlation observables with accuracy O(M-1 + t epsilon(2)), for correlation time t where epsilon(2) is related to the variance of mean value approximation h. Furthermore, the proof derives a precise asymptotic representation of the Weyl symbol of the Gibbs density operator using a path integral formulation. Numerical experiments on a model problem with one nuclei and two electron states show that the mean-field dynamics has similar or better accuracy than standard molecular dynamics based on the ground state electron eigenvalue.
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| 4. |
- Kammonen, Aku, 1984-, et al.
(författare)
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Adaptive random fourier features with metropolis sampling
- 2019
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Ingår i: Foundations of Data Science. - : American Institute of Mathematical Sciences. - 2639-8001. ; 0:0, s. 0-0
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Tidskriftsartikel (refereegranskat)abstract
- The supervised learning problem todetermine a neural network approximation $\mathbb{R}^d\ni x\mapsto\sum_{k=1}^K\hat\beta_k e^{{\mathrm{i}}\omega_k\cdot x}$with one hidden layer is studied asa random Fourier features algorithm. The Fourier features, i.e., the frequencies $\omega_k\in\mathbb{R}^d$,are sampled using an adaptive Metropolis sampler.The Metropolis test accepts proposal frequencies $\omega_k'$, having corresponding amplitudes $\hat\beta_k'$, with the probability$\min\big\{1, (|\hat\beta_k'|/|\hat\beta_k|)^\gamma\big\}$,for a certain positive parameter $\gamma$, determined by minimizing the approximation error for given computational work.This adaptive, non-parametric stochastic method leads asymptotically, as $K\to\infty$, to equidistributed amplitudes $|\hat\beta_k|$, analogous to deterministic adaptive algorithms for differential equations. The equidistributed amplitudes are shown to asymptotically correspond to the optimal density for independent samples in random Fourier features methods.Numerical evidence is provided in order to demonstrate the approximation properties and efficiency of the proposed algorithm. The algorithm is testedboth on synthetic data and a real-world high-dimensional benchmark.
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| 5. |
- Kammonen, Aku, 1984-, et al.
(författare)
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Canonical quantum observables for molecular systems approximated by ab initio molecular dynamics
- 2018
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Ingår i: Annales Henri Poincaré. - : Springer Nature. - 1424-0637 .- 1424-0661. ; 19, s. 2727-2781
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Tidskriftsartikel (refereegranskat)abstract
- It is known that ab initio molecular dynamics based on the electron ground state eigenvaluecan be used to approximate quantum observables in the canonical ensemble when the temperature is low compared tothe first electron eigenvalue gap. This work proves that a certain weighted average of the different ab initio dynamics, corresponding to each electron eigenvalue, approximates quantum observables for any temperature.The proof uses the semi-classical Weyl law to show thatcanonical quantum observables of nuclei-electron systems, based on matrix valued Hamiltonian symbols, can be approximated by ab initio molecular dynamics with the error proportional to the electron-nuclei mass ratio. The resultincludes observables that depend on correlations in time. A combination of the Hilbert-Schmidt inner product for quantum operators and Weyl's lawshows that the error estimate holds %for observables and Hamiltonian symbols that have three and five bounded derivatives, respectively, provided the electron eigenvalues are distinct for any nuclei positionand the observables are in diagonal form with respect to the electron eigenstates.
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| 6. |
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| 7. |
- Kammonen, Aku, 1984-, et al.
(författare)
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Canonical quantum observables for molecular systems approximated by ab inition molecular dynamics
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- It is known that ab initio molecular dynamics based on the electron ground state eigenvaluecan be used to approximate quantum observables in the canonical ensemble when the temperature is low compared tothe first electron eigenvalue gap. This work proves that a certain weighted average of the different ab initio dynamics, corresponding to each electron eigenvalue, approximates quantum observables for any temperature.The proof uses the semi-classical Weyl law to show thatcanonical quantum observables of nuclei-electron systems, based on matrix valued Hamiltonian symbols, can be approximated by ab initio molecular dynamics with the error proportional to the electron-nuclei mass ratio. The resultincludes observables that depend on correlations in time. A combination of the Hilbert-Schmidt inner product for quantum operators and Weyl's lawshows that the error estimate holds %for observables and Hamiltonian symbols that have three and five bounded derivatives, respectively, provided the electron eigenvalues are distinct for any nuclei positionand the observables are in diagonal form with respect to the electron eigenstates.
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| 8. |
- Kammonen, Aku, 1984-, et al.
(författare)
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COMPUTATIONAL ALGORITHMS FOR CANONICAL ENSEMBLE OBSERVABLES
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We consider canonical ensembles of molecular systems. We propose four numerical algorithms for efficient computation of the canonical ensemble molecular dynamics observables. The four algorithms can each be efficient in different situations. For example in low temperatures we can make use of the fact that the lowest electron energy levels contributes most to the observable.
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| 9. |
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| 10. |
- Kammonen, Aku, 1984-, et al.
(författare)
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SMALLER GENERALIZATION ERROR DERIVED FOR DEEP COMPARED TO SHALLOW RESIDUAL NEURAL NETWORKS
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Estimates of the generalization error are proved for a residual neural network with $L$ random Fourier features layers $\bar z_{\ell+1}=\bar z_\ell + \mathrm{Re}\sum_{k=1}^K\bar b_{\ell k}e^{\mathrm{i}\omega_{\ell k}\bar z_\ell}+\mathrm{Re}\sum_{k=1}^K\bar c_{\ell k}e^{\mathrm{i}\omega'_{\ell k}\cdot x}$. An optimal distribution for the frequencies $(\omega_{\ell k},\omega'_{\ell k})$ of the random Fourier features $e^{\mathrm{i}\omega_{\ell k}\bar z_\ell}$ and $e^{\mathrm{i}\omega'_{\ell k}\cdot x}$ is derived. This derivation is based on the corresponding generalization error for the approximation of the function values $f(x)$. The generalization error turns out to be smaller than the estimate ${\|\hat f\|^2_{L^1(\mathbb{R}^d)}}/{(LK)}$ of the generalization error for random Fourier features with one hidden layer and the same total number of nodes $LK$, in the case the $L^\infty$-norm of $f$ is much less than the $L^1$-norm of its Fourier transform $\hat f$. This understanding of an optimal distribution for random features is used to construct a new training method for a deep residual network that shows promising results.
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