Lecture Notes: Deep Operator Networks with George Karniadakis

Introduction

• Speaker: George Karniadakis
• Fields: Applied Mathematician
• Interests: Stochastic differential equations, computational fluid dynamics, machine learning for scientific applications
• Education: Mechanical Engineering (National Technical University in Athens, MIT for Master's and Ph.D.)
• Career: Held positions at Princeton, currently at Brown University
• Current Focus: Deep Operator Networks (DeepONet)

DeepONet Overview

AI Crossroads and GPT-3 Example

• Current state of AI: Potential stagnation point
• Example: GPT-3 by OpenAI
  • 175 billion parameters, training cost of $5 million
  • Despite scale, still makes mistakes
  • Scaling up isn't the only solution
  • Requires higher-level abstraction and non-linear operators

Efficiency in Mathematically Intelligent Robots

• Teaching calculus to robots is inefficient
• High computational requirements
• Energetic considerations: The human brain is highly efficient

Universal Theorem of Function Approximation

Traditional Neural Networks

• Focus on function approximation
• Example: Image classification
• Mapping from finite-dimensional space to finite-dimensional space

Higher-Level Approximation

• Functionals and non-linear operators
• Mapping from infinite-dimensional space (function space) to infinite-dimensional space
• Operators can include derivates, integrals, differential equations, biological systems

Learning Non-Linear Operators

Generalization

• Need for extrapolating outside the training distribution
• Example: Classification problem with generalization error quantified differently (e.g., by data distribution and network smoothness)

Probability of Neighborhood and Self/Mutual Cover

• Data distribution concepts like the probability of neighborhood
• Introducing self cover and mutual cover for different classes
• Relation between data distribution (self and mutual cover) and network smoothness

Physics-Informed Neural Networks (PINNs)

Combining Physics and Data

• Regularizes the neural network with physical laws (conservation of mass, momentum, energy)
• Addresses the data scarcity in scientific problems

Example Applications

• Simple problem: Solving an ordinary differential equation (ODE) both within and outside the training domain
• Hidden Fluid Mechanics: Use auxiliary data (e.g., from smoke or thermal gradients) to infer pressure and velocity fields
• Biomedical Example: Modeling brain aneurysms

DeepONet: Problem Setup and Practical Implications

Concept Overview

• Map a function from a compact space to an operator's output
• Use neural networks to approximate the input space (branch network) and output space (trunk network)

Generalization Examples

• Universal approximation for functions and operators
• Approximation accuracy improves with better representation of the input space

Applications

• Integral operator approximation
• Non-linear operator cases
• Real-time PDE solutions using pre-trained networks

Special Cases and Advanced Topics

Fractional Calculus

• Captures memory effects and anomalous transport
• Learning fractional derivatives with neural networks

Stochastic Differential Equations

• Handling colored noise via Karhunen-Loève expansion

Hypersonic Flows

• Prediction of trajectories and handling shocks/discontinuities
• Use of pre-trained neural networks for rapid predictions

Final Thoughts

• Exponential convergence in certain problems
• Future work: High-level abstraction for complex multi-physics and multi-scale problems

Questions and Answers

• Discussion on why neural networks outperform traditional methods in some contexts
• Specifics on training, error bounds, and application to convolutional neural networks

Note: This summary captures key points of the lecture and can be augmented with additional details as needed for further study.