Abstract
The outline of this thesis is the application of Machine Learning (ML) techniques to
solving the Time Independent Schrödinger Equation (TISE). The solving of the TISE is an
explicitly non-polynomially (NP) hard computational problem that suffers from the curse
of dimensionality. The motivation of this thesis is to see if the computational techniques
of ML, which have solved previously intractable problems, are applicable for the NP hard
problems of quantum mechanics. If a polynomially scaling algorithm that could solve
the TISE accurately existed, it would revolutionise computational modelling of quantum
mechanical problems allowing for efficient drug design, simulating super heavy nuclei,
and much more.
In the work of this thesis, we begin with a proof-of-principle exercise of solving a variety
of Hamiltonians with the use of Deep Neural Networks (DNNs) as the ansatz of the
wavefunction. We exploit the use of the Universal Approximation Theorem (UAT), which
in principle allows any function to be approximated to an arbitrary precision, to effectively
learn the ground-state wavefunction of a Hamiltonian directly via Variational Monte
Carlo (VMC) techniques. This work can be seen as an extension to VMC with DNNs
which is an emerging field in computation modelling of the Schrödinger equation. The
main limitation of VMC is the functional form of the wavefunction, by representing the
wavefunction as a DNN allows the UAT to be exploited to efficiently solve the TISE.
In summary, the ML-based approach of this thesis shows strong evidence of efficient
wavefunction representation via DNN ansätze with results that compare favourably with
exact diagonalisation. This has the implication of DNNs being an efficient approximation
to exact numerical methodologies without suffering from the explicit NP hardness of
exact diagonalisation.