PDE-Driven Methods for Scientific Discovery: Dislocation Simulation, Protein Design, and Neural PDE Solvers

Abstract

Partial differential equations (PDEs) serve as the cornerstone of mathematical modeling across scientific disciplines, from predicting material defects to simulating biomolecular interactions, leading to transformative innovations in computational science. This thesis advances these frontiers by addressing three challenges in materials science, drug discovery, and neural operator solvers.

In the first part, we present a continuum formulation for the dislocation climb velocity based on the densities of dislocations. The obtained continuum formulation is an accurate approximation of the Green’s function based discrete dislocation dynamics method. The continuum dislocation climb formulation has the advantage of accounting for both the long-range effects of vacancy diffusion and that of the Peach-Koehler climb force, and the two long-range effects are canceled into a short-range effect (integral with a fast-decaying kernel) and in some special cases, a completely local effect. This significantly simplifies the calculation in the Green’s function-based discrete dislocation dynamics method, in which a linear system has to be solved over the entire system for the long-range effect of vacancy diffusion, and the long-range Peach-Koehler climb force has to be calculated. This obtained continuum dislocation climb velocity can be applied in any available continuum dislocation dynamics framework. We also present numerical validations for this continuum climb velocity and simulation examples for implementation in continuum dislocation dynamics frameworks.

In the second part, we propose Poisson Flow-based AntiBody Generator (PF-ABGen), a novel antibody structure and sequence designer. We adopt the protein structure representation with torsion and bond angles, which allows us to represent the conformations more elegantly and take advantage of the efficient sampling procedure of the Poisson Flow Generative Model. Our computational experiments demonstrate that PF-ABGen can generate natural and realistic antibodies in an efficient and reliable way. In particular, PF-ABGen can also be applied to the design of antibodies with variable lengths.

In the third part, we present a robust neural operator framework that enhances stability through adversarial training while preserving accuracy. We formulate operator learning as a min-max optimization problem, where the model is trained against worst-case input perturbations to achieve consistent performance under both normal and adversarial conditions. We demonstrate that our method not only achieves good performance on standard inputs but also maintains high fidelity under adversarial perturbed inputs. The results highlight the importance of stability-aware training in operator learning and provide a foundation for developing reliable neural PDE solvers in real-world applications, where input noise and uncertainties are inevitable.

Date of Award	2025
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology
Supervisor	Yang XIANG (Supervisor)