Hamidreza Hashempoorikderi

This project explores three complementary directions for solving physical systems. Physical systems appear across science and engineering, often described by PDEs, ODEs, or more complex equations. Accurately solving these equations remains a central challenge in applied mathematics and computational science.

Classical theoretical approaches provide rigorous tools for analyzing existence, uniqueness, and stability of solutions. These rely on function spaces, operator theory, and weak formulations. Techniques such as Galerkin methods, variational principles, and spectral theory exemplify this tradition. Modern neural operator models extend these ideas into machine learning. DeepONet and Fourier Neural Operators learn mappings between function spaces rather than pointwise predictions. Recent extensions add memory mechanisms, such as state-space models, to capture temporal dependencies. Probabilistic operator learning further generalizes this by modeling distributions of solutions, with diffusion-based methods showing promise.

A second paradigm integrates physics directly into neural networks. Physics-Informed Neural Networks (PINNs) incorporate governing equations into the loss function during training. This constrains models to obey known physics while fitting data. Extensions such as XPINNs and SPINNs improve scalability by domain decomposition or variable separation. Despite progress, challenges remain in applying PINNs to high-dimensional or strongly nonlinear systems. This active area offers opportunities to refine methods for more complex real-world problems. A third and emerging paradigm leverages large language models (LLMs). Trained on vast corpora of mathematics and science, LLMs show potential for symbolic reasoning and solver generation. They can assist in formulating equations, boundary conditions, or symbolic regression tasks. Although still in its early stages, LLM-based approaches open a novel path for modeling and reasoning about physical systems.

Ultimately, this PhD project will focus on inferring physical systems while drawing on all three of these directions.