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- Kammonen, Aku, 1984-, et al.
(author)
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Adaptive random fourier features with metropolis sampling
- 2019
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In: Foundations of Data Science. - : American Institute of Mathematical Sciences. - 2639-8001. ; 0:0, s. 0-0
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Journal article (peer-reviewed)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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| 2. |
- Kammonen, Aku, 1984-, et al.
(author)
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Canonical quantum observables for molecular systems approximated by ab initio molecular dynamics
- 2018
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In: Annales Henri Poincaré. - : Springer Nature. - 1424-0637 .- 1424-0661. ; 19, s. 2727-2781
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Journal article (peer-reviewed)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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| 3. |
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| 4. |
- Plechác, P., et al.
(author)
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The classical limit of quantum observables in the conservation laws of fluid dynamics
- 2019
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In: Communications in Mathematical Sciences. - : International Press of Boston, Inc.. - 1539-6746 .- 1945-0796. ; 17:8, s. 2191-2221
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Journal article (peer-reviewed)abstract
- In the classical work by Irving and Zwanzig [J.H. Irving and R.W. Zwanzig, J. Chem. Phys., 19, 1173-1180, 1951] it has been shown that quantum observables for macroscopic density, momentum and energy satisfy the conservation laws of fluid dynamics. In this work we derive the corresponding classical molecular dynamics limit by extending Irving and Zwanzig's result to matrix-valued potentials for a general quantum particle system. The matrix formulation provides the classical limit of the quantum observables in the conservation laws also in the case where the temperature is large compared to the electron eigenvalue gaps. The classical limit of the quantum observables in the conservation laws is useful in order to determine the constitutive relations for the stress tensor and the heat flux by molecular dynamics simulations. The main new steps to obtain the molecular dynamics limit are: (i) to approximate the dynamics of quantum observables accurately by classical dynamics, by diagonalizing the Hamiltonian using a nonlinear eigenvalue problem, (ii) to define the local energy density by partitioning a general potential, applying perturbation analysis of the electron eigenvalue problem, (iii) to determine the molecular dynamics stress tensor and heat flux in the case of several excited electron states, and (iv) to construct the initial particle phase-space density as a local grand canonical quantum ensemble determined by the initial conservation variables.
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