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shock-capturing-neural-operators
GitHub针对含激波、间断或陡峭梯度的PDE（如低粘Burgers方程），解决标准FNO的吉布斯振荡问题。提供ShockFNO局部-全局架构及非周期边界反射填充技术，提升离散解精度。
Trigger Scenarios
PDE包含激波或接触间断
标准FNO出现吉布斯振荡
处理低粘度Burgers或可压缩Euler方程
需要非周期边界条件
Install
npx skills add synthetic-sciences/openscience --skill shock-capturing-neural-operators -g -y
SKILL.md
Frontmatter
{
"name": "shock-capturing-neural-operators",
"tags": [
"Neural Operator",
"FNO",
"Shocks",
"Gibbs",
"Discontinuity",
"Boundary Conditions",
"Spectral"
],
"author": "Synthetic Sciences",
"license": "MIT",
"version": "1.0.0",
"category": "physics",
"description": "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.",
"dependencies": [
"torch>=2.1.0",
"numpy>=1.24.0"
]
}
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).
Solution 1: Local-Global Architecture (ShockFNO)
Add a parallel local convolution branch alongside the spectral path. Local conv captures sharp features without spectral artifacts.
class FNOBlock(nn.Module):
"""Gated local-global spectral block."""
def __init__(self, width, modes, local_kernel=7):
super().__init__()
self.spectral = SpectralConv1d(width, width, modes) # Global (FFT)
self.pointwise = nn.Conv1d(width, width, 1) # Bias path
self.local_conv = nn.Conv1d(width, width, local_kernel,
padding=local_kernel//2) # Local (shock-scale)
self.gate = nn.Parameter(torch.tensor(0.3)) # Learned balance
def forward(self, x):
global_out = self.spectral(x) + self.pointwise(x)
local_out = self.local_conv(x)
alpha = torch.sigmoid(self.gate)
return (1 - alpha) * global_out + alpha * local_out
Design choices:
local_kernel=7: Covers ~7 grid cells. For 1024-pt grid with shock width ~1 cell, this captures the shock + immediate neighborhood.gateinitialized at 0.3 (sigmoid → ~0.57): starts slightly favoring global, learns to balance per-layer.- In trained models, gate values converge to 0.3-0.6 across layers — both branches contribute.
Inspired by: LOGLO-FNO (arXiv:2504.04260), but simplified: standard Conv1d instead of local spectral convolutions, scalar gate instead of spatial attention.
Solution 2: Reflection Padding for Non-Periodic BCs
Standard FNO uses FFT which assumes periodicity. For outgoing/transmissive BCs, waves that exit the domain re-enter from the opposite side in Fourier space.
class ShockTubeFNO1d(nn.Module):
def __init__(self, ..., pad_size=32):
self.pad_size = pad_size
# ... standard FNO layers ...
def forward(self, x, grid):
x = self.fc0(x).permute(0, 2, 1) # [B, W, X]
# REFLECTION PADDING before spectral layers
x = F.pad(x, [self.pad_size, self.pad_size], mode='reflect')
for block in self.blocks:
x = block(x)
# Remove padding after spectral layers
x = x[:, :, self.pad_size:-self.pad_size]
return self.projection(x)
Why reflection padding works:
- Creates a smooth continuation at boundaries (unlike zero-padding which introduces a jump)
- Reduces spectral leakage from boundary discontinuities
pad_size=32on 256-point grid = ~12.5% extension — substantial
Comparison:
- Standard FNO: 2-point zero padding (0.2% extension) — nearly useless
- Our approach: 32-point reflection (12.5%) — meaningful improvement
- Optimal: Fourier Continuation (FC-PINO, arXiv:2211.15960) — polynomial continuation to periodic domain
Solution 3: Resolution Scaling
For shocks, resolution is the single biggest factor. Shock width ≈ ν/U for viscous problems.
| Viscosity ν | Shock Width | Min Grid Points | Recommendation |
|---|---|---|---|
| 0.1 | ~0.1 | ~100 | 256 is fine |
| 0.01 | ~0.01 | ~1000 | 512 minimum, 1024 better |
| 0.001 | ~0.001 | ~10000 | 1024 (full native), needs A100 |
| inviscid | 0 (true discontinuity) | ∞ | As high as possible; 256-1024 |
Memory scaling: Doubling resolution roughly doubles memory and compute per epoch.
Frequency-Band Error Analysis
Diagnostic tool: decompose prediction error by wavenumber to identify where FNO fails.
def frequency_band_errors(pred, target, bands={"low": (0, 5), "mid": (5, 13), "high": (13, None)}):
"""Compute relative spectral error per frequency band."""
pred_ft = torch.fft.rfft(pred, dim=1)
tgt_ft = torch.fft.rfft(target, dim=1)
n_modes = pred_ft.shape[1]
errors = {}
for name, (lo, hi) in bands.items():
hi = hi or n_modes
err = torch.abs(pred_ft[:, lo:hi] - tgt_ft[:, lo:hi]).mean().item()
nrm = torch.abs(tgt_ft[:, lo:hi]).mean().item() + 1e-20
errors[name] = {"abs": err, "rel": err / nrm}
return errors
Typical results for shock problems:
- Low band (k=0-4): rel error ~0.1% — excellent (large-scale structure captured)
- Mid band (k=5-12): rel error ~5-10% — moderate (shock-scale features)
- High band (k≥13): rel error ~30% — worst (Gibbs-dominated)
This analysis reveals the fundamental spectral limit and motivates wavelet-based approaches.
Architecture Selection for Different Shock Problems
| Problem | Architecture | Key Features |
|---|---|---|
| Moderate viscosity (ν=0.01-0.1) | Enhanced FNO | More modes (32), H1 loss, noise injection |
| Low viscosity (ν≤0.001) | ShockFNO | Local conv branch, freq loss, full resolution |
| Multi-variable shocks (Euler, NS) | MultiFNO + ShockFNO | Per-channel normalization + local conv |
| Non-periodic BCs (outgoing) | ShockTubeFNO | Reflection padding + boundary loss |
| Riemann problems (shock tube) | ShockTubeFNO | All of the above |
Future Directions (from literature)
| Approach | Reference | Promise |
|---|---|---|
| Wavelet Neural Operator (WNO) | — | Wavelets naturally represent discontinuities |
| Fourier Continuation (FC-PINO) | arXiv:2211.15960 | Principled non-periodic extension |
| Convolutional Neural Operator (CNO) | NeurIPS 2023 | No spectral assumption at all |
| Godunov loss functions | arXiv:2405.11674 | Entropy-satisfying shock capture |
| DCT/DST replacement for FFT | arXiv:2507.21757 | Non-periodic spectral methods |
Version History
- e9844a4 Current 2026-07-11 17:32
Dependencies
-
required
torch>=2.1.0 -
required
numpy>=1.24.0
