added file

Author: mhjensen

doc/src/TimeEvolution/projectPMM.tex

@@ -2,7 +2,7 @@
 \usepackage{graphicx} % Required for inserting images
 \usepackage{hyperref}
 
-\title{Machine learning and artificial intelligence for quantum-mechanical systems}
+\title{Machine learning and non-linear dynamics}
 \author{Master of Science thesis project}
 \date{November 2024}
 
@@ -13,83 +13,58 @@
 \section{Scientific aims}
 
 
-The aim of this master of science project is to develop many-body
-theories for studies of strongly interacting quantum mechanical
-many-particle systems using novel methods from deep learning theories,
-in particular advanced neural networks and other generative models.
-
-For interacting many-particle systems where the degrees of freedom
-increase exponentially, quantum mechanical many-body methods like
-quantum Monte Carlo methods, Coupled Cluster theory, Green's function
-theories, density functional theories and other, play a central role
-in understanding experiments in a wide range of fields, spanning from
-atomic and molecular physics and thereby quantum chemistry to
-condensed matter physics, materials science, nano-technologies and
-quantum technologies and finally processes like fusion and fission in
-nuclear physics. This list is definitely not exhaustive as the are
-many other areas of applications for quantum mechanical many-body
-theories.
-
-
-Quantum Monte Carlo techniques are widely applicable and have been
-used in studies of a large range of systems.  The main difficulty with
-QMC calculations of fermionic systems is ensuring the fermionic
-antisymmetry is respected. In Diffusion Monte Carlo calculations,
-which in principle yield the exact solutions in the limit of lon
-simulation times, the prescription to ensure this is fixing the nodes
-of the system to prevent the large errors that would come with the
-summation of alternating signs. Unfortunately this prescription is not
-variational in nature and can in some cases result in convergence to
-energies lower than the true ground state of the system. This is one
-of the advantages of VMC calculations since they are known to be
-variational the resulting wavefunction will always have an energy
-larger than or equal to the true ground state wavefunction.
-
-The problem with variational Monte Carlo calculations has been the
-choice of trial wave function.  Recently, several research groups have
-introduced, with great success, neural networks as a way to represent
-the trial wave function. Recent works on infinite nuclear
-matter\cite{us2023a,us2024}, the unitary Fermi gas \cite{us2023b}, and
-Daniel Haas Beccatini's recent master of science thesis project
-\cite{daniel2024} have shown that one can obtain results of equal
-accuratness as the theoretical benchmark calculations provided by
-diffusion Monte Carlo results.  This has opened up a new area of
-research and the present thesis project aims at developing further
-deep learning approaches to the studies of strongly interacting
-many-body systems.
-
-The plans here are to extend the studies in \cite{daniel2024} to
-studies of low-dimensional systems such as quantum dots and the
-infinite electron gas in two dimensions. These are systems of great
-interest for materials science studies, nano-technologies and quantum
-technologies. A proper understanding of the properties of such systems
-will play a crucial role in designing for example quantum gates and
-circuits.  These systems are studied experimentally at the university
-of Oslo at the Center for Materials Science and Nanotechnologies
-(SMN). The theoretical activity at the Center for Computing in Science
-Education and the Computational Physics research group have through
-the last years developed a strong collaboration with several
-researchers at the SMN.
-
-The thesis of Daniel Haas Beccatini \cite{daniel2024} with codes and
-additional material will serve as an excellent backgroun material.
+The aim of this master of science project is to study the solution of
+time-dependent differential equations such as the time-dependent
+Schr\"odinger equation in quantum mechanics using a newly develop
+machine learning method, the so-called Parametric Matrix Model (PMM)
+approach \cite{us2024}. This method has been shown to be surprisingly
+stable and subccesful and performs better (less paramters) in many
+cases than most of the standard deep learning methods for both
+regression and classification problems.  The PMM has been shown to
+have a great potential in solving in particular non-linear dynamics
+and time-dependent problems.
+
+
+One of the first steps in solving any physics problem is identifying
+the governing equations.  While the solutions to those equations may
+exhibit highly complex phenomena, the equations themselves have a
+simple and logical structure.  The structure of the underlying
+equations leads to important and nontrivial constraints on the
+analytic properties of the solutions.  Some well-known examples
+include symmetries, conserved quantities, causality, and analyticity.
+Unfortunately, these constraints are usually not reproduced by machine
+learning algorithms, leading to inefficiencies and limited accuracy
+for scientific computing applications. Current physics-inspired and
+physics-informed machine learning approaches aim to constrain the
+solutions by penalizing violations of the underlying equations, but
+this does not guarantee exact adherence.  Furthermore, the results of
+many modern deep learning methods suffer from a lack of
+interpretability.  To address these issues, we introduced recently a
+new class of machine learning algorithms called parametric matrix
+models \cite{us2024}.  Parametric matrix models (PMMs) use matrix
+equations similar to the physics of quantum systems and learn the
+governing equations that lead to the desired outputs.  Since quantum
+mechanics provides a fundamental microscopic description of all
+physical phenomena, the existence of such matrix equations is clear.
+Parametric matrix models take the additional step of applying the
+principles of model order reduction and reduced basis methods to find
+efficient approximate matrix equations with finite dimensions.
+
 
 The plan for this thesis project is as follows:
 \begin{itemize}
-\item Spring 2025: follow courses and get familair with Monte Carlo methods
-and how to use neural networks to solve simpler quantum
-mechanical. The software developed in \cite{daniel2024} can serve as a
-guidance for developing own code.
-\item Fall 2025: Include stochastic resampling \cite{daniel2024} and develop code for studies of both bosonic and fermionic systems.
-\item Spring 2026: Apply formalism and code to studies of two-dimensional systems like quantum dots and/or the infinite electron gas in two dimensions. Finalize thesis by May 2026.
+\item Spring 2025: follow courses and get familiar with the PMM method
+and reproduce several of the test examples discussed in \cite{us2024}. 
+The notebook at \cite{dannynotebook} can serve as a useful start for getting to know the method.
+\item Fall 2025: The main application of the method is to time-dependent quantum mechanical problems. The thesis starts with time-dependent Hartree-Fock theory as discussed in the work of Zanghellini {\em et al.}, see \cite{zanghellini} applied to a system of electrons confined in one-dimensional traps. The aim is to apply the PMM to the study of such dynamical systems. Codes developed by us for the time-dependent Hartree-Fock method will be provided.
+\item Spring 2026: With a working code for the time-dependent Hartree-Fock method, the next step is to extend this to more advanced many-body methods such as the time-dependent multiconfiguration Hartree-Fock method. The first step is to reproduce the results of Zanghellini {\em et al.}, see \cite{zanghellini} before applying the formalism and codes to two-dimensional systems of quantum dots. These systems are highly relevant candidates for making quantum components such as various gates. The latter can be used in studies of the time-evolution of entanglement. We expect that the thesis will finalized by May 2026. 
 \end{itemize}
 
 \begin{thebibliography}{99}
 
-\bibitem{us2023a} Bryce Fore, Jane M. Kim, Giuseppe Carleo, Morten Hjorth-Jensen, Alessandro Lovato, and Maria Piarulli, Dilute neutron star matter from neural-network quantum states, Physical Review Research {\bf 5}, 033062 (2023) and \url{https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.033062}
-\bibitem{us2023b} Jane Kim, Gabriel Pescia, Bryce Fore, Jannes Nys, Giuseppe Carleo, Stefano Gandolfi, Morten Hjorth-Jensen, Alessandro Lovato, Neural-network quantum states for ultra-cold Fermi gases, Nature Communications Physics {\bf 7}, 148 (2024) and \url{https://www.nature.com/articles/s42005-024-01613-w}
-\bibitem{us2024} Bryce Fore, Jane Kim, Morten Hjorth-Jensen, Alessandro Lovato, Investigating the crust of neutron stars with neural-network quantum states, Nature Communications Physics in press  and \url{https://arxiv.org/abs/2407.21207}
-\bibitem{daniel2024} Daniel Haas Beccatini, Master of Science thesis, University of Oslo, 2024, Deep Learning Methods for Quantum Many-body Systems, A study on Neural Quantum States, \url{https://www.duo.uio.no/handle/10852/113984}
+\bibitem{us2024} Patrick Cook, Danny Jammooa, Morten Hjorth-Jensen, Daniel D. Lee, Dean Lee, Parametric Matrix Models, \url{https://arxiv.org/abs/2401.11694}
+\bibitem{dannynotebook} Danny Jammoa, notebook at  \url{https://github.com/dannyjammooa/Parametric-Matrix-Models/tree/main}  
+\bibitem{zanghellini} Jürgen Zanghellini et al, J. Phys. B: At. Mol. Opt. Phys. {\bf 37},  763 (2004) and \url{https://iopscience.iop.org/article/10.1088/0953-4075/37/4/004/pdf}
 \end{thebibliography}  
 
 
@@ -99,4 +74,3 @@ \section{Scientific aims}
 
 
 
-