Forcing and Diagnosing Failure Modes of Fourier Neural Operators Across Diverse PDE Families

Fourier Neural Operators (FNOs) have shown strong performance in learning solution maps of partial differential equations (PDEs), but their robustness under distribution shifts, long-horizon rollouts, and structural perturbations remains poorly understood. We present a systematic stress-testing framework that probes failure modes of FNOs across five qualitatively different PDE families: dispersive, elliptic, multi-scale fluid, financial, and chaotic systems. Rather than optimizing in-distribution accuracy, we design controlled stress tests - including parameter shifts, boundary or terminal condition changes, resolution extrapolation with spectral analysis, and iterative rollouts - to expose vulnerabilities such as spectral bias, compounding integration errors, and overfitting to restricted boundary regimes. Our large-scale evaluation (1,000 trained models) reveals that distribution shifts in parameters or boundary conditions can inflate errors by more than an order of magnitude, while resolution changes primarily concentrate error in high-frequency modes. Input perturbations generally do not amplify error, though worst-case scenarios (e.g., localized Poisson perturbations) remain challenging. These findings provide a comparative failure-mode atlas and actionable insights for improving robustness in operator learning.

💡 Research Summary

This paper conducts a comprehensive stress‑testing study of Fourier Neural Operators (FNOs), focusing on their robustness when faced with out‑of‑distribution (OOD) scenarios that are common in real scientific and engineering applications. Five qualitatively distinct partial differential equation (PDE) families are selected to span a broad spectrum of dynamics: (1) the nonlinear Schrödinger equation (NLS), a dispersive complex‑valued wave equation; (2) the Poisson equation, a canonical elliptic problem dominated by boundary conditions; (3) the 2‑D incompressible Navier–Stokes equations in vorticity form, a multi‑scale fluid system sensitive to viscosity and forcing; (4) the Black‑Scholes equation, a one‑dimensional parabolic PDE from quantitative finance with a terminal payoff condition; and (5) the Kuramoto–Sivashinsky (K‑S) equation, a chaotic fourth‑order PDE.

For each PDE, the authors train 200 independent FNO models (totaling 1,000 models) using a common architecture (≈10⁵ parameters, 16 Fourier modes) and a uniform training pipeline. The training data are generated from an in‑distribution regime specific to each PDE (e.g., moderate amplitude initial conditions for NLS, a fixed range of viscosity for Navier–Stokes, smooth European‑call payoffs for Black‑Scholes). After achieving baseline relative L₂ errors of a few percent, the models are subjected to five systematic OOD stress tests, each designed to probe a different failure mechanism:

A. Parameter/coefficients shift – key physical parameters (nonlinearity κ, viscosity ν, volatility σ, coefficient field a(x)) are moved beyond the training range.
B. Boundary/terminal condition shift – the models are evaluated on boundary data or payoff functions that differ substantially from those seen during training.
C. Resolution extrapolation with spectral analysis – models trained on a coarse grid (e.g., 64×64) are applied to a finer grid (128×128); error fields are decomposed into Fourier modes to assess spectral bias.
D. Long‑horizon rollout stability – for time‑dependent PDEs, the learned operator is iteratively composed to simulate long‑time evolution, measuring error growth and defining a rollout degradation factor.
E. Input perturbation sensitivity – small random perturbations (≈5 % of the input L₂ norm) are added to inputs; the ratio of perturbed‑to‑unperturbed errors quantifies Lipschitz‑type stability.

The authors report a degradation factor D = (OOD error) / (baseline error) for each stressor, averaging across seeds and providing 95 % confidence intervals. Key findings include:

1. Parameter shifts cause the largest error blow‑up. For NLS, increasing κ beyond the training range yields D≈3.9; for Poisson, roughening a(x) leads to D≈10.2; for Black‑Scholes, raising σ results in D≈4.5. Even though the parameter is supplied as an explicit input, the FNO fails to extrapolate the nonlinear mapping.

2. Boundary/terminal condition shifts also degrade performance dramatically. Switching Poisson’s Dirichlet patterns or using a digital (discontinuous) option payoff in Black‑Scholes produces D≈5–6, indicating that the network has largely memorized the training family of boundary data rather than learning the underlying operator.

3. Resolution extrapolation shows modest average error increase (D≈1.0–1.1) but a pronounced spectral bias. Error energy concentrates in the highest Fourier modes that were absent from the training bandwidth, confirming that FNOs act as band‑limited reconstructions and cannot generate new high‑frequency content on finer grids.

4. Long‑horizon rollouts expose compounding integration errors. Even with low one‑step errors (≈2 %), repeated application over 50 steps leads to D≈7–9 for NLS and K‑S, and D≈5.2 for Navier–Stokes at low viscosity, reflecting the interaction of numerical error with intrinsic dynamical instability.

5. Input perturbation sensitivity is generally low (D≈1.0–1.2), but Poisson exhibits heightened sensitivity (D≈2.1) to localized coefficient spikes, highlighting a specific vulnerability to high‑frequency input variations.

From these observations, the authors draw several design recommendations: (i) broaden training distributions to include diverse parameter ranges and boundary families; (ii) incorporate multi‑scale or hierarchical Fourier representations to alleviate spectral bias; (iii) embed physics‑based stability constraints (e.g., energy conservation, Lyapunov regularization) to curb rollout divergence; and (iv) employ adversarial or localized perturbation augmentation to improve robustness against sharp input changes.

In summary, the paper delivers a “failure‑mode atlas” for Fourier Neural Operators across a spectrum of PDEs, revealing that while FNOs achieve impressive in‑distribution accuracy, they are prone to severe degradation under realistic distribution shifts, especially in parameter extrapolation, boundary condition changes, and long‑time integration. The systematic methodology and extensive empirical evidence provide a valuable benchmark for future work aiming to build more robust, generalizable neural operators for scientific computing.