Neural network iterative diagonalization method to solve eigenvalue problems in quantum mechanics
We propose a multi-layer feed-forward neural network iterative diagonalization method (NNiDM) to compute some eigenvalues and eigenvectors of large sparse complex symmetric or Hermitian matrices. The NNiDM algorithm is developed by using the complex (or real) guided spectral transform Lanczos (cGSTL...
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| Published in | Physical chemistry chemical physics : PCCP Vol. 17; no. 21; pp. 1471 - 1482 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
England
07.06.2015
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| Subjects | |
| Online Access | Get full text |
| ISSN | 1463-9076 1463-9084 1463-9084 |
| DOI | 10.1039/c5cp01438g |
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| Abstract | We propose a multi-layer feed-forward neural network iterative diagonalization method (NNiDM) to compute some eigenvalues and eigenvectors of large sparse complex symmetric or Hermitian matrices. The NNiDM algorithm is developed by using the complex (or real) guided spectral transform Lanczos (cGSTL) method, thick restart technique, and multi-layered basis contraction scheme. Artificial neurons (or nodes) are defined by a set of formally orthogonal Lanczos polynomials, where the biases and weights are dynamically determined through a series of cGSTL iterations and small matrix diagonalizations. The algorithm starts with one random vector. The last output layer produces wanted eigenvalues and eigenvectors near a given reference value
via
a linear transform diagonalization approach. Since the algorithm uses the spectral transform technique, it is capable of computing interior eigenstates in dense spectrum regions. The general NNiDM algorithm is applied for calculating energies, widths, and wavefunctions of two typical molecules HO
2
and CH
4
as examples.
The neural network iterative diagonalization structure for computing the eigenstates of complex symmetric or Hermitian matrices. |
|---|---|
| AbstractList | We propose a multi-layer feed-forward neural network iterative diagonalization method (NNiDM) to compute some eigenvalues and eigenvectors of large sparse complex symmetric or Hermitian matrices. The NNiDM algorithm is developed by using the complex (or real) guided spectral transform Lanczos (cGSTL) method, thick restart technique, and multi-layered basis contraction scheme. Artificial neurons (or nodes) are defined by a set of formally orthogonal Lanczos polynomials, where the biases and weights are dynamically determined through a series of cGSTL iterations and small matrix diagonalizations. The algorithm starts with one random vector. The last output layer produces wanted eigenvalues and eigenvectors near a given reference value
via
a linear transform diagonalization approach. Since the algorithm uses the spectral transform technique, it is capable of computing interior eigenstates in dense spectrum regions. The general NNiDM algorithm is applied for calculating energies, widths, and wavefunctions of two typical molecules HO
2
and CH
4
as examples.
The neural network iterative diagonalization structure for computing the eigenstates of complex symmetric or Hermitian matrices. We propose a multi-layer feed-forward neural network iterative diagonalization method (NNiDM) to compute some eigenvalues and eigenvectors of large sparse complex symmetric or Hermitian matrices. The NNiDM algorithm is developed by using the complex (or real) guided spectral transform Lanczos (cGSTL) method, thick restart technique, and multi-layered basis contraction scheme. Artificial neurons (or nodes) are defined by a set of formally orthogonal Lanczos polynomials, where the biases and weights are dynamically determined through a series of cGSTL iterations and small matrix diagonalizations. The algorithm starts with one random vector. The last output layer produces wanted eigenvalues and eigenvectors near a given reference value via a linear transform diagonalization approach. Since the algorithm uses the spectral transform technique, it is capable of computing interior eigenstates in dense spectrum regions. The general NNiDM algorithm is applied for calculating energies, widths, and wavefunctions of two typical molecules HO2 and CH4 as examples.We propose a multi-layer feed-forward neural network iterative diagonalization method (NNiDM) to compute some eigenvalues and eigenvectors of large sparse complex symmetric or Hermitian matrices. The NNiDM algorithm is developed by using the complex (or real) guided spectral transform Lanczos (cGSTL) method, thick restart technique, and multi-layered basis contraction scheme. Artificial neurons (or nodes) are defined by a set of formally orthogonal Lanczos polynomials, where the biases and weights are dynamically determined through a series of cGSTL iterations and small matrix diagonalizations. The algorithm starts with one random vector. The last output layer produces wanted eigenvalues and eigenvectors near a given reference value via a linear transform diagonalization approach. Since the algorithm uses the spectral transform technique, it is capable of computing interior eigenstates in dense spectrum regions. The general NNiDM algorithm is applied for calculating energies, widths, and wavefunctions of two typical molecules HO2 and CH4 as examples. We propose a multi-layer feed-forward neural network iterative diagonalization method (NNiDM) to compute some eigenvalues and eigenvectors of large sparse complex symmetric or Hermitian matrices. The NNiDM algorithm is developed by using the complex (or real) guided spectral transform Lanczos (cGSTL) method, thick restart technique, and multi-layered basis contraction scheme. Artificial neurons (or nodes) are defined by a set of formally orthogonal Lanczos polynomials, where the biases and weights are dynamically determined through a series of cGSTL iterations and small matrix diagonalizations. The algorithm starts with one random vector. The last output layer produces wanted eigenvalues and eigenvectors near a given reference value via a linear transform diagonalization approach. Since the algorithm uses the spectral transform technique, it is capable of computing interior eigenstates in dense spectrum regions. The general NNiDM algorithm is applied for calculating energies, widths, and wavefunctions of two typical molecules HO sub(2) and CH sub(4) as examples. We propose a multi-layer feed-forward neural network iterative diagonalization method (NNiDM) to compute some eigenvalues and eigenvectors of large sparse complex symmetric or Hermitian matrices. The NNiDM algorithm is developed by using the complex (or real) guided spectral transform Lanczos (cGSTL) method, thick restart technique, and multi-layered basis contraction scheme. Artificial neurons (or nodes) are defined by a set of formally orthogonal Lanczos polynomials, where the biases and weights are dynamically determined through a series of cGSTL iterations and small matrix diagonalizations. The algorithm starts with one random vector. The last output layer produces wanted eigenvalues and eigenvectors near a given reference value via a linear transform diagonalization approach. Since the algorithm uses the spectral transform technique, it is capable of computing interior eigenstates in dense spectrum regions. The general NNiDM algorithm is applied for calculating energies, widths, and wavefunctions of two typical molecules HO 2 and CH 4 as examples. We propose a multi-layer feed-forward neural network iterative diagonalization method (NNiDM) to compute some eigenvalues and eigenvectors of large sparse complex symmetric or Hermitian matrices. The NNiDM algorithm is developed by using the complex (or real) guided spectral transform Lanczos (cGSTL) method, thick restart technique, and multi-layered basis contraction scheme. Artificial neurons (or nodes) are defined by a set of formally orthogonal Lanczos polynomials, where the biases and weights are dynamically determined through a series of cGSTL iterations and small matrix diagonalizations. The algorithm starts with one random vector. The last output layer produces wanted eigenvalues and eigenvectors near a given reference value via a linear transform diagonalization approach. Since the algorithm uses the spectral transform technique, it is capable of computing interior eigenstates in dense spectrum regions. The general NNiDM algorithm is applied for calculating energies, widths, and wavefunctions of two typical molecules HO2 and CH4 as examples. |
| Author | Yu, Hua-Gen |
| AuthorAffiliation | Department of Chemistry Brookhaven National Laboratory |
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| Author_xml | – sequence: 1 givenname: Hua-Gen surname: Yu fullname: Yu, Hua-Gen |
| BackLink | https://www.ncbi.nlm.nih.gov/pubmed/25959361$$D View this record in MEDLINE/PubMed |
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| CitedBy_id | crossref_primary_10_1021_acs_jpca_9b02295 crossref_primary_10_1063_1_4921411 crossref_primary_10_1016_j_cherd_2017_12_029 crossref_primary_10_1063_1_4961642 crossref_primary_10_1007_s11467_023_1325_z |
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| Title | Neural network iterative diagonalization method to solve eigenvalue problems in quantum mechanics |
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