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In low-mass X-ray binaries, the accretion of stellar material onto a neutron star can fuel unstable thermonuclear flashes known as Type I X-ray bursts. Simulating these events using computational models can provide valuable information about the nature of the accreting system. One-dimensional (1D) astrophysics codes with large nuclear reaction networks are the current state-of-the-art for simulating X-ray bursts. These codes can track the evolution of isotopes through thousands of nuclear reaction pathways, to predict the released nuclear energy and final composition of the ashes. In this thesis, I make extensive use of KEPLER, a 1D code at the forefront of these efforts. I first present improvements to the setup and analysis of KEPLER burst models. By accounting for nuclear heating in the initial conditions, I shorten the thermal burn-in time, thereby reducing computational expense and producing more consistent burst trains. To model bursts fueled by transient accretion events, I perform the first such simulations with fully time-dependent accretion rates. Building upon previous efforts to model the "Clocked Burster", GS 1826$-$238, I precompute a grid of 3840 simulations and sample the interpolated results using Markov Chain Monte Carlo (MCMC) methods. By comparing the predictions to multi-epoch observations, I obtain posterior probability distributions for the system parameters. I then extend these MCMC methods to the pure-helium burster, 4U 1820$-$30, using a grid of 168 simulations. Finally, I discuss potential improvements for future studies, to further develop the computational modelling of accreting neutron stars.