Artificial Neural Networks to Solve Inverse Problems in Quantum Physics

Sammanfattning: 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.