DeepONet for solving nonlinear partial differential equations with physics-informed training.

Area of Science:

• Machine Learning
• Applied Mathematics
• Numerical Analysis

Background:

• Traditional methods require separate neural networks for each nonlinear partial differential equation (PDE).
• Operator learning offers a generalized approach to solving PDEs without retraining.
• Deep learning models are increasingly applied to scientific problems, necessitating robust theoretical underpinnings.

Purpose of the Study:

• Investigate DeepONet, a specific operator learning model, for solving nonlinear PDEs.
• Analyze the approximation capabilities of DeepONet's branch and trunk networks in physics-informed training.
• Derive theoretical bounds for the generalization error of DeepONet in Sobolev norms.

Main Methods:

• Physics-informed neural networks (PINNs) and operator learning framework.
• DeepONet architecture with deep branch and simple trunk networks.
• Rademacher complexity and pseudo-dimension analysis for error bound derivation.

Main Results:

• Complex branch networks significantly improve DeepONet's performance.
• Simpler trunk networks demonstrate optimal effectiveness.
• A rigorous bound on DeepONet's generalization error for nonlinear PDEs was derived.

Conclusions:

• DeepONet shows promise for generalized PDE solving via operator learning.
• The study provides crucial theoretical error estimates for physics-informed machine learning.
• This work bridges a gap in understanding the generalization capabilities of operator learning models.