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.
• gate initialized 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=32 on 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