Analysis of neural-network-based PDE-solving algorithms

Thesis

Neural networks are powerful function approximation tools in scientific computing. Thanks to recent advances in the modern electronic engineering industry, computational resources are more accessible than ever to train neural networks to perform various tasks. In recent years, algorithms that numerically solve partial differential equations (PDEs) using neural networks have become popular, as these methods can overcome the curse of dimensionality, which prevents classical methods from solving high-dimensional equations.

Despite their practical success, the mathematical properties of these algorithms have not yet been well-studied. Compared to other typical neural network training tasks, solving PDEs requires matching boundary conditions and PDE operators simultaneously. The unboundedness of PDE operators (in L²) also makes the analysis even harder.

In this thesis, we aim to provide a rigorous mathematical analysis of related algorithms in several aspects. A new algorithm (Q-PDE) for solving PDEs is proposed, and a few numerical examples are given. The training process of the neural network approximator is characterized, for which we also prove that an infinite-dimensional ODE models the limiting behavior. A novel neural tangent kernel is described, which drives the ODE system. By applying proper functional analysis tools, we can show the convergence of neural networks to PDE solutions in the limit regime. In addition, we model the dynamics of policy iteration methods with neural networks that solve PDEs originating from stochastic control problems by reformulating them as a sequence of linear PDEs.

Authors: Jiang, D

• DOI: 10.5287/ora-xrjobazjy
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English