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shock-capturing-neural-operators
Architectures and techniques for neural operators on discontinuous PDE solutions (shocks, contact discontinuities, steep gradients). Covers local-global spectral design (ShockFNO), reflection padding for non-periodic BCs, resolution scaling for shock width, and frequency-band error diagnostics. Use for low-viscosity Burgers, compressible Euler, Riemann problems, or any PDE where standard FNO produces Gibbs oscillations.
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# Shock-Capturing Neural Operators
## When to Use
- PDE solutions with shocks, contact discontinuities, or steep gradients
- Low-viscosity Burgers, compressible Euler, Riemann problems
- Non-periodic boundary conditions (outgoing, transmissive, Dirichlet)
- Any problem where standard FNO produces Gibbs-like oscillations
## The Fundamental Problem: Gibbs Phenomenon in FNO
Standard FNO uses FFT → truncate modes → iFFT. For discontinuous functions, Fourier coefficients decay as O(1/k), producing oscillatory artifacts near discontinuities regardless of mode count. This is the Gibbs phenomenon — a mathematical limitation, not a training problem.
**Impact:** FNO nRMSE degrades 10× going from smooth to shock problems (e.g., Burgers ν=0.1: 2.9e-3 vs ν=0.001: 2.9e-2).
