DeepONet (Deep Operator Network) is a universal function approximator for operators — a neural network architecture capable of learning the mapping between infinite-dimensional function spaces (e.g., mapping initial conditions to the solution of a PDE over time).

What Is DeepONet?

• Structure: Two sub-networks (Branch and Trunk).
• Branch Net: Encodes the input function $u(x)$ at fixed sensors.
• Trunk Net: Encodes the coordinates $(y)$ where we want to evaluate the output.
• Output: The dot product of Branch and Trunk outputs gives the value of the operator $G(u)(y)$.
• Theorem: Universal Approximation Theorem for Operators (Chen & Chen, 1995).

Why It Matters

• Real-Time Physics: Can predict the outcome of a simulation (e.g., airflow over a wing) in milliseconds instead of hours.
• Data-Driven: Learns the physics from data without needing to know the governing equations explicitly.
• Generalization: Works for any resolution or grid size.

DeepONet is the "MLP" of operator learning — the foundational architecture for scientific machine learning tasks involving differential equations.