Adaptive Hypernetwork-Based Parameter Estimation for Efficient PINN Training

Description:

Physics-Informed Neural Networks (PINNs) provide a powerful framework for solving partial differential equations (PDEs) by embedding physics constraints into the loss function. However, their performance is highly sensitive to network initialization and hyperparameter selection. Traditional PINNs require extensive tuning, and their training can be inefficient when applied to varying PDE conditions.

This thesis aims to explore the use of hypernetworks for adaptive parameter estimation in PINNs, leveraging tabular foundation models to predict optimal network weights dynamically. The focus will be on three key directions:

• Bounded Hyperparameter Prediction: Utilizing lower and upper bounds to constrain hypernetwork-generated parameters for stability and interpretability.
• Task-Aware Parameter Estimation: Investigating whether a tabular foundation model can generalize across PDE variations and improve initialization strategies.
• Meta-Learned Adaptation: Exploring whether pretraining on diverse PDEs enables fast adaptation to new problems with few-shot learning.

Level: 

Masters Thesis

Requirements:

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

Research Questions:

• Can a tabular foundation model improve hypernetwork-based parameter estimation for PINNs?
• Does bounding the predicted parameters enhance stability and convergence?
• Can meta-learning accelerate adaptation to new PDEs with minimal data?
  

References:

• Torres E, Schiefer J, Niepert M. Adaptive Physics-informed Neural Networks: A Survey. arXiv preprint arXiv:2503.18181. 2025 Mar 23.
• Mueller AC, Curino CA, Ramakrishnan R. MotherNet: Fast Training and Inference via Hyper-Network Transformers. InNeurIPS 2024 Third Table Representation Learning Workshop.
• 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).