Approximation rates of DeepONets for learning operators arising from advection-diffusion equations

Area of Science:

• Numerical analysis
• Machine learning
• Partial differential equations

Background:

• Operator learning approximates complex functions using neural networks.
• Previous work by Chen and Chen (1995) and Lu et al. (2021) established foundational approximation rates.
• Advection-diffusion equations are fundamental in modeling various physical phenomena.

Purpose of the Study:

• To analyze the approximation rates of operator learning for solving linear and nonlinear advection-diffusion equations.
• To investigate how network architecture and data smoothness affect learning performance.
• To extend the understanding of operator learning beyond simple function approximation.

Main Methods:

• Analysis of approximation rates using theoretical frameworks.
• Application of operator learning, specifically sum of products of branch and trunk networks.
• Consideration of advection-diffusion equations with and without reaction terms.

Main Results:

• The study quantifies approximation rates for learning solution operators.
• Demonstrated dependence of these rates on the architecture of branch networks.
• Identified the smoothness of input and output data as a critical factor influencing approximation accuracy.

Conclusions:

• Approximation rates in operator learning are sensitive to network design, particularly branch networks.
• Data characteristics, specifically input-output smoothness, significantly impact the efficiency of learning solution operators.
• This research provides insights into optimizing operator learning for solving differential equations.