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Fourier Neural Operators for Parametric PDEs

Background for reading Li et al., "Fourier Neural Operator for Parametric Partial Differential Equations" (arXiv:2010.08895). Sentence-by-sentence breakdown of operator learning, the neural operator architecture, and the Fourier-space parameterization, with concrete Darcy-flow examples and SVG diagrams throughout.

Prerequisites: Multivariable calculus, basic PDEs (heat/wave equation), familiarity with Fourier transforms, some exposure to neural networks. No operator theory background assumed — everything is built from scratch.
Part I — The Problem & Architecture
01

The Operator Learning Problem

What problem are we solving? The spatial domain D, function spaces A and U, the operator G† that maps functions to functions, the parametric approximation, and discretization with resolution invariance. Grounded in 2D Darcy flow throughout.

Sections 1–6 · Paper §2 · Darcy flow · Resolution invariance
02

The Neural Operator Architecture

The lift–iterate–project pipeline. The lifting operator P, the iterative kernel integral layer (Definition 1), the kernel integral operator (Definition 2), the projection Q, and the full forward pass with concrete dimensions.

Sections 7–12 · Paper §3 · Definitions 1–2 · O(n²) bottleneck
Part II — The Fourier Insight
03

From Kernels to Fourier Space

Brief Fourier refresher, the convolution theorem, translation-invariant kernels, the key insight (Definition 3): learn R directly in Fourier space. Concrete tensor shapes and mode truncation.

Sections 13–17 · Paper §4 (first half) · Definition 3 · Mode truncation
04

The Complete FNO Layer & Forward Pass

The full Fourier layer equation, two parallel paths (global Fourier + local W), the discrete FFT implementation, a complete forward-pass walkthrough with dimensions, and complexity analysis.

Sections 18–22 · Paper §4 (second half) · FFT algorithm · O(n log n)
Part III — Applications
05

FNO on Real PDEs

Burgers' equation (1D), Darcy flow (2D) with full experimental results, Navier–Stokes (2D+time), zero-shot super-resolution, and an honest assessment of what FNO can and can't do.

Sections 23–27 · Paper §5 · Benchmarks · FNO vs PINNs vs DeepONet
Part IV — Application: Ultrasonic NDT
06

Elastic Waves & Ultrasonic NDT

The elastic wave equation, P-waves and S-waves, Snell's law and mode conversion, the angle beam NDT setup, cracks as scatterers, and casting NDT as an operator learning problem.

Sections 28–32 · Elastic waves · COMSOL setup · NDT operator
07

FNO for Crack Detection

Crack parameterization, FNO-3D vs signal-level architectures, training data generation from COMSOL, evaluation metrics, and the challenges of applying FNO to high-frequency wave scattering.

Sections 33–37 · Architecture design · Training pipeline · FNO variants