Physics-Informed Correction for Long-Rollout Stabilization in Neural Operators

Description:

Neural operators have shown promise in learning solution operators for complex PDEs, enabling fast inference for unseen conditions. However, these models often suffer from error accumulation in long rollouts, leading to degradation in solution quality. Traditional stabilization methods rely on explicit regularization or retraining, which can be computationally expensive.

This thesis aims to explore a Physics-Informed Neural Network (PINN)-based corrective mechanism that dynamically adjusts neural operator predictions when errors become significant. The key directions include:

• Residual-Based Error Monitoring: Computing the PDE residual at regular intervals to detect instability in the predicted solution.
• Adaptive PINN Correction: If the residual exceeds a predefined threshold, then perform rapid fine-tuning of the neural operator parameters using a PINN loss to restore physical consistency.
• Few-Step Optimization for Stabilization: Investigating whether minimal parameter updates with physics-informed losses can efficiently correct long-rollout degradation without full retraining.

Level:

Masters Thesis

Requirements:

• Python & JAX.
• Experience with Neural Operators (Fourier Neural Operators, DeepONets).
• Machine Learning (PINNs, Meta-Learning, Transfer Learning).
  

Research Questions:

• Can a PINN-based correction mechanism effectively mitigate error accumulation in long rollouts?
• How frequently should residual-based corrections be applied to balance efficiency and accuracy?
• Can a few-step optimization using a physics-informed loss restore solution stability without excessive computational cost?
  

References:

• Li Z, Zheng H, Kovachki N, Jin D, Chen H, Liu B, Azizzadenesheli K, Anandkumar A. Physics-informed neural operator for learning partial differential equations. ACM/JMS Journal of Data Science. 2024 May 20;1(3):1-27.
• Kovachki NB, Li Z, Liu B, Azizzadenesheli K, Anandkumar A, Stuart AM. Neural operator: Learning maps between function spaces. J. Machine Learning Research, 2023.
• Brandstetter J, Worrall D, Welling M. Message passing neural PDE solvers. ICML 2022.