Enhancing Physics-Informed Neural Networks via Residual-Guided Optimization and Parallel Subdomain Training

Description:

Physics-Informed Neural Networks (PINNs) have demonstrated advantages over traditional PDE solvers, such as ease of implementation, the ability to incorporate partial data, and built-in heuristics for solution validation (e.g., residual maps and loss functions). However, PINNs suffer from slow convergence and may struggle to find accurate solutions for complex PDEs.

This thesis aims to explore three key directions to improve PINN efficiency:

1. Residual-Guided Optimization: Leveraging residual maps to refine the optimization process dynamically.
2. Subdomain-Based Partial Training: Investigating whether training/fine-tuning on high-error regions can accelerate convergence.
   
3. Parallelized Subdomain Optimization: Exploring whether network parameters can be updated independently across subdomains to enable parallel training.

Level:

Masters Thesis

Requirements:

• Python & JAX
• General knowledge of PDEs
• Machine Learning (Transfer Learning, Meta-Learning)

Research Questions:

• Can residual-based parameter sampling outperform a learnable hypernetwork for weight prediction?
• Can residual maps guide selective fine-tuning to improve convergence speed?
• Can PINN training be parallelized via subdomain optimization?

References:

• Torres E, Schiefer J, Niepert M. Adaptive Physics-informed Neural Networks: A Survey. arXiv preprint arXiv:2503.18181. 2025 Mar 23.
• Xiao Y, Yang LM, Du YJ, Song YX, Shu C. Radial basis function-differential quadrature-based physics-informed neural network for steady incompressible flows. Physics of Fluids. 2023 Jul 1;35(7).
• Ling L, Kansa EJ. Preconditioning for radial basis functions with domain decomposition methods. Mathematical and Computer modeling. 2004 Dec 1;40(13):1413-27.