Shallow Machine Learning Architectures for Enhancing Computational Modelling in Physics and Engineering

Supervisors: Christoforos Rekatsinas, George Giannakopoulos, Panagiotis Krokidas

Description:

This thesis proposes a data-efficient machine learning framework for enhancing computational modelling in physics and engineering, with applications ranging from quantum mechanics and molecular dynamics to nonlinear mechanics. The work focuses on three main methodologies: Physics-Informed Neural Networks (PINNs), Optimization of a Discrete Loss (ODIL), and Neural Operators.
PINNs will be used to embed governing physical equations directly into the learning process, allowing the model to solve forward and inverse problems with limited data. This is particularly relevant for problems governed by differential equations, such as quantum wave equations, transport phenomena, and nonlinear mechanical response.
ODIL will be investigated as an alternative physics-based learning strategy, where the governing equations are discretized first and the loss function is constructed directly from the residuals of the discrete numerical problem. This approach can improve numerical stability and provide a closer link between machine learning and classical computational solvers.
Neural Operators will be explored for learning mappings between functions, such as material parameters, boundary conditions, forcing terms, or potentials, and the corresponding physical response. This makes them suitable for accelerating repeated simulations in molecular dynamics, quantum mechanics, and nonlinear multiphysics problems.
The thesis will develop and compare these approaches on representative benchmark problems, starting from simplified quantum or molecular systems and extending toward nonlinear mechanics applications. Emphasis will be placed on data efficiency, physical consistency, computational acceleration, and the ability of the models to generalize across different conditions. The final objective is to assess how physics-informed and operator-learning methods can support faster and more reliable computational workflows for complex physics and engineering problems.

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Qualifications required: Python programming; Machine Learning; basic knowledge of numerical methods and scientific computing