DeepOnet - Learning Nonlinear Operators Based on the Universal Approximation Theorem of Operators

Explore a comprehensive lecture on DeepOnet, a novel neural network architecture designed to learn nonlinear operators based on the universal approximation theorem. Delve into the theoretical foundations, practical applications, and unique features of DeepOnet as presented by George Karniadakis from Brown University. Discover how this innovative approach leverages the power of neural networks to approximate complex continuous operators and systems with high accuracy. Examine various examples, including explicit operators like integrals and fractional Laplacians, as well as implicit operators representing deterministic and stochastic differential equations. Investigate the network's structure, consisting of branch and trunk networks, and its ability to encode discrete input function spaces and output function domains. Analyze the impact of different input function space formulations on generalization error and explore potential applications in fields such as fluid mechanics and brain aneurysm modeling. Gain insights into advanced concepts like autonomy, hidden fluid mechanics, and multiphysics simulations, while considering future research directions and potential improvements to the DeepOnet architecture.

Introduction
Universal approximation theorem
Why is it different
Classification problem
New concepts
Theorem
Smoothness
What is a pin
Autonomy
Hidden Fluid Mechanics
Espresso
Brain Aneurysm
Operators
Problem setup
The universal approximation theorem
Crossproduct
Deep Neural Network
Input Space
Recap
Example
Results
Learning fractional operators
Individual trajectories
Nonlinearity
Multiphysics
Eminem
Spectral Methods
Can we bound the error in term of the operator norm
Can we move away from compactness assumption
What allows these networks to approximate exact solutions
Can it learn complex userdefined operators
Wavelets instead of sigmoids
Variational pins
Comparing to real neurons
How to test this idea