Overcoming Computational Barriers through Neural Network PDE Solvers: Theory, Methodology and Application

Abstract

This thesis addresses three fundamental challenges in traditional numerical methods for solving partial differential equations (PDEs) through strategic integration with deep learning techniques.

First, for nonlinear time-evolution PDEs, we introduce a neural hybrid solver that uses a convolutional neural network trained by a novel implicit-scheme-informed learning strategy to predict initial guesses for Newton’s method. The proposed approach significantly reduces iteration counts and achieves acceleration while preserving the structure of solutions. Theoretical analysis of how initialization affects the iteration count is provided. Additionally, we/ analyze the generalization error of the proposed unsupervised learning strategy in both fully-discrete and semi-discrete settings.

Second, to overcome the Kolmogorov barrier in multiscale kinetic transport equations, we develop reduced order models (ROMs) to predict solutions under the parametric setting. We first develop a piecewise linear ROM by introducing a novel goal-oriented adaptive time partitioning strategy with the aid of a coarsening strategy. Next, for problems where a local linear approximation is not sufficiently efficient, we further develop a hybrid ROM, which strategically applies autoencoder-based nonlinear ROMs only when necessary. Numerical experiments show that the proposed ROMs successfully predict full-order solutions at unseen parameter values with both efficiency and accuracy.

Finally, we introduce the SSFMnet, a novel neural operator designed for solving a general class of nonlinear time-evolution PDEs based on the split-step Fourier method (SSFM). Originally developed for nonlinear Schrödinger equations, we extend the SSFM to address a wider range of nonlinear equations. The core architecture adapts this generalized numerical scheme by employing Fourier integral operators to approximate the Fourier multipliers within the linear and nonlinear solution operators. Structured following the AutoFlow framework mimicking the evolution of equations, SSFMnet overcomes the limitations of previous AutoFlow-based PDE solvers, which (1) require analytical solutions for nonlinear terms and (2) lack continuous-discrete equivalence.

Numerical experiments confirm that the proposed methods demonstrate significant improvements in efficiency, accuracy, and robustness compared to classical PDE solvers.

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