Skip to main content

LUP Student Papers

LUND UNIVERSITY LIBRARIES

Artificial Neural Networks to Solve Inverse Problems in Quantum Physics

Lantz, Victor LU (2019) PHYM01 20191
Mathematical Physics
Department of Physics
Solid State Physics
Abstract
Inverse problems are important in quantum physics as their solutions are essential in order to describe
a number of systems using measurable information, e.g. excitation energies or material properties. The
problem with inverse problems is that they are usually very hard to solve. One method that could be useful
in solving these problems is artificial neural networks. Artificial neural networks have received a lot of
attention lately as they have shown great results in solving many difficult problems e.g. super-resolution in
imaging [1], [2]. However, their use in the field of physics have been limited so far.
In this master’s thesis, neural networks have been applied to a few inverse quantum mechanical problems to
see if it is... (More)
Inverse problems are important in quantum physics as their solutions are essential in order to describe
a number of systems using measurable information, e.g. excitation energies or material properties. The
problem with inverse problems is that they are usually very hard to solve. One method that could be useful
in solving these problems is artificial neural networks. Artificial neural networks have received a lot of
attention lately as they have shown great results in solving many difficult problems e.g. super-resolution in
imaging [1], [2]. However, their use in the field of physics have been limited so far.
In this master’s thesis, neural networks have been applied to a few inverse quantum mechanical problems to
see if it is possible for them to solve these problems. The inverse problems that are in focus in this project
are: solving the external potential of quantum mechanical systems using either the eigenvalue spectrum
or the density function. Lastly, the task of going from potential to density function is also treated. In the
project the problems were limited to their 1D form with computationally generated data that made use of a
finite difference method.
All of the problems investigated in the thesis have been successfully solved using dense networks. The
inverse problem where the potential was computed using the eigenvalue spectrum was solved using a custom
error function. This error function accounted for the 1D potentials having the same eigenvalue spectrum
when reversed along spatial axis. It was shown that the network predictions were very close the the actual
potentials. Both the density to potential problem and its reverse problem were solved as well. These
results showed not only good predictions, but also that the networks were able to generalise well to particle
numbers they had not trained on.
Based on the results, it has been shown that it is possible to solve these problems using artificial neural
networks. Now that this has been shown, the next step would be to apply this method to real data in order
to create solution tools that could have practical use for real problems. Additionally, it is likely that there
are other problems in the field where neural networks could prove useful. (Less)
Popular Abstract (Swedish)
I detta arbete har det visats att artificiella neuronnät är lämpliga för att lösa svåra inversproblem inom
kvantfysik. Relativt enkla program har använts för att räkna ut kvantmekaniska potentialer från antingen
deras energispektrum, eller hur partiklar fördelar sig i dem. Vilket visar på att denna teknik skulle kunna
vara lämplig för andra svårare problem i området. Det skulle också kunna användas för att enklare bestämma
egenskaper hos nya material och läkemedel endast genom att mäta energispektrum eller partikelfördeling.
Please use this url to cite or link to this publication:
author
Lantz, Victor LU
supervisor
organization
course
PHYM01 20191
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Artificial Neural Networks, ANN, Deep learning, Quantum physics, Inverse problem, Inverse, Machine learning, Density functional theory, DFT, Physics, Computational physics
language
English
id
8986347
date added to LUP
2019-06-20 10:53:52
date last changed
2020-04-23 15:53:31
@misc{8986347,
  abstract     = {{Inverse problems are important in quantum physics as their solutions are essential in order to describe
a number of systems using measurable information, e.g. excitation energies or material properties. The
problem with inverse problems is that they are usually very hard to solve. One method that could be useful
in solving these problems is artificial neural networks. Artificial neural networks have received a lot of
attention lately as they have shown great results in solving many difficult problems e.g. super-resolution in
imaging [1], [2]. However, their use in the field of physics have been limited so far.
In this master’s thesis, neural networks have been applied to a few inverse quantum mechanical problems to
see if it is possible for them to solve these problems. The inverse problems that are in focus in this project
are: solving the external potential of quantum mechanical systems using either the eigenvalue spectrum
or the density function. Lastly, the task of going from potential to density function is also treated. In the
project the problems were limited to their 1D form with computationally generated data that made use of a
finite difference method.
All of the problems investigated in the thesis have been successfully solved using dense networks. The
inverse problem where the potential was computed using the eigenvalue spectrum was solved using a custom
error function. This error function accounted for the 1D potentials having the same eigenvalue spectrum
when reversed along spatial axis. It was shown that the network predictions were very close the the actual
potentials. Both the density to potential problem and its reverse problem were solved as well. These
results showed not only good predictions, but also that the networks were able to generalise well to particle
numbers they had not trained on.
Based on the results, it has been shown that it is possible to solve these problems using artificial neural
networks. Now that this has been shown, the next step would be to apply this method to real data in order
to create solution tools that could have practical use for real problems. Additionally, it is likely that there
are other problems in the field where neural networks could prove useful.}},
  author       = {{Lantz, Victor}},
  language     = {{eng}},
  note         = {{Student Paper}},
  title        = {{Artificial Neural Networks to Solve Inverse Problems in Quantum Physics}},
  year         = {{2019}},
}