Lectures

Date Topics Readings
9/18 Week 1 Course Introduction [slides] [recording]
  • Course syllabus and requirements
  • AI in Science and Engineering
  • Mathematical modeling with PDEs
  • Computational challenges
  • Motivation for AI approaches
9/25 Week 2 Introduction to Deep Learning [slides] [recording]
  • Introduction to using deep learning to model physical systems governed by PDEs.
  • Structure of MLPs with layers, weights, biases, and activation functions (sigmoid, tanh, ReLU, etc.).
  • Gradient descent, stochastic gradient descent (SGD), mini-batch SGD
  • Motivation for convolutional neural networks (CNNs) to handle high-dimensional inputs efficiently.
10/2 Week 3 Introduction to Physics-Informed Neural Networks [slides] [recording]
  • Introduction to Physics-Informed Neural Networks (PINNs).
  • Extending PINNs to reconstruct unknown solutions or parameters from partial measurements.
  • PINNs unify data-driven learning and physics-based modeling, offering flexible, mesh-free solvers for forward and inverse PDE problems.
10/9 Week 4 PINNs - Theoretical insights [slides] [recording]
  • Review of Physics-Informed Neural Networks (PINNs) for solving PDEs.
  • Theoretical analysis of PINN error - relation between training error, PDE residuals, and total approximation error.
  • Conditions ensuring convergence - coercivity, quadrature approximation, and DNN expressivity.
  • Rigorous error bounds for linear and nonlinear PDEs (Kolmogorov, Black-Scholes, Navier-Stokes, Burgers' equation).
  • Gradient descent dynamics and conditioning in PINN training; interpretation via NTK and preconditioning.
  • Practical performance and challenges:successes on smooth PDEs, difficulties on shocks or high-conditioning problems.
  • Overview of acceleration and stabilization techniques (causal learning, hard BCs, multi-stage networks).
10/16 Week 5 Operator Learning - Introduction [slides] [recording]
  • Transition from physics-informed learning to data-driven approaches.
  • Introduction to operator learning
  • Parametric PDE learning:deep networks approximating observables for low-dimensional parameterizations.
  • Operator learning:approximating infinite-dimensional mappings from data distributions.
  • Neural Operators generalizing DNNs to function spaces.
  • Fourier Neural Operators (FNOs):convolution in Fourier space, efficient and translation-invariant.
  • Theoretical foundation:universal approximation theorems for FNOs.
  • Practical challenges:bridging continuous operators and discrete numerical data (continuous-discrete equivalence).
10/23 Week 6 Operator Learning - FNO [slides] [recording]
  • Continuation of Operator Learning - FNO.
  • Continuous-discrete equivalence and Representation equivalent neural operators (ReNOs).
  • Why CNN and FNO are not ReNO.
10/30 Week 7 Operator Learning - ReNO [slides] [recording]
  • Continuation of Operator Learning - ReNO.
  • Convolutional Neural Operator (CNO), which is constructed to be a ReNO
11/6
11/13
11/20
11/27
12/4
12/11
12/18

Tutorials

Date Topics Exercise Solutions
9/22 Function Approximation with Pytorch [exercise] [solutions]
9/25 Cross Validation and Intro to CNNs [exercise] [solutions]
10/6 PINN Training [exercise] [solutions]
10/13 Fourier Neural Operator [exercise] [solutions]
10/20 No Tutorial This Week.
10/27 Convolutional Neural Operator [exercise]
11/3
11/10
11/17
11/24
12/1
12/8
12/15
12/22