| 9/17 |
Week 1 Course Introduction [slides] [video]
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Course logistics: no exam, project-based grading, tutorials as the core programming component
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Scope: physics and engineering (majority of the course) plus a short module on chemistry and biology
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Scientific method as PDE modeling: from natural phenomena to partial differential equations
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PDEs as a compact language of nature and a strong prior across physics, engineering, and beyond
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Why PDEs are hard to solve: few closed-form solutions and costly numerical methods
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Many-query tasks and the sim-to-real gap as motivation for combining PDE priors with data-driven AI
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| 9/24 |
Week 2 Introduction to Deep Learning [slides] [video]
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Learn PDE-modeled physics from data with neural networks
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From McCulloch–Pitts and the perceptron to MLPs: affine maps, pointwise activations, and stacked layers
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Universal approximation: an MLP can represent a continuous map, but the parameters still have to be found
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Supervised learning: labeled data, training loss, and a validation set to check generalization
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Training by gradient descent, backpropagation, and mini-batch SGD
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Beyond dense MLPs: CNNs and other structured architectures for high-dimensional inputs
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| 10/1 |
Week 3 Introduction to Physics-Informed Neural Networks 1 [slides] [video]
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PINNs solve a PDE without labeled data by minimizing interior, initial, and boundary residuals
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Residuals use automatic differentiation and are trained on collocation points
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Easy to implement, and high-dimensional heat and finance examples show only a mild growth of error with dimension
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Forward problems include viscous conservation laws, incompressible Navier–Stokes, and dispersive PDEs such as KdV
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A small amount of data can help, and the same loss handles data assimilation from partial measurements
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The framework also covers radiative transfer, including inverse parameter identification
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| 10/8 |
Week 4 PINNs — Theoretical Insights [slides] [video]
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Where PINNs succeed: high-dimensional PDEs, parametric problems, and inverse problems
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Total error is controlled by the PDE residual when the PDE is coercive and the network and quadrature are accurate enough
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Explicit bounds for Kolmogorov equations, Navier–Stokes, and viscous Burgers
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Gradient descent on the physics-informed loss, and why high-frequency Poisson and advection are badly conditioned
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Preconditioning can rebalance data and physics terms, but shocks and stiff problems remain difficult
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| 10/15 |
Week 5 Operator Learning — Introduction [slides] [video]
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From solving one PDE with PINNs to learning the map from inputs to solutions
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Parametric PDE learning works when the input has a low-dimensional parameterization, such as airfoil shape
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Discretize-then-learn with MLPs or CNNs runs into model size and resolution limits
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Neural operators learn the continuous map first; FNOs do this with convolutions in Fourier space
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FNOs have a universal approximation theorem, but continuous operators and discrete training data do not automatically match
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| 10/22 |
Week 6 Operator Learning — FNO, ReNO, and CNO [slides] [video]
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Continuous–discrete equivalence: discretize-then-learn and learn-then-discretize should agree across grids
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A representation-equivalent neural operator keeps the same map when the grid changes
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CNNs are not representation equivalent, because the kernel is fixed on a discrete grid
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FNOs are not representation equivalent either: pointwise activations can leave the bandlimit
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Convolutional neural operators use continuous convolutions on bandlimited functions and are representation equivalent by construction
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