# Diagnosing Failure Modes of Neural Operators Across Diverse PDE Families

arXiv: https://arxiv.org/abs/2601.11428 · HTML: https://mlsanity.com/paper/2601.11428/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 38 references that are in the map.
- First posted on arXiv 16 Jan 2026.

## Builds on (most cited first, 26 of 26)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Domain Generalization: A Survey](https://mlsanity.com/paper/2103.02503/) (2021)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [APEBench: A Benchmark for Autoregressive Neural Emulators of PDEs](https://mlsanity.com/paper/2411.00180/) (2024)
- [On the training and generalization of deep operator networks](https://mlsanity.com/paper/2309.01020/) (2023)
- [Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities](https://mlsanity.com/paper/2210.01074/) (2022)
- [Discretization Error of Fourier Neural Operators](https://mlsanity.com/paper/2405.02221/) (2024)
- [Bounding the Rademacher Complexity of Fourier neural operators](https://mlsanity.com/paper/2209.05150/) (2022)
- [Fourier Neural Operators Explained: A Practical Perspective](https://mlsanity.com/paper/2512.01421/) (2025)
- [Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspective](https://mlsanity.com/paper/2310.06379/) (2023)
- [Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Laws](https://mlsanity.com/paper/2401.01783/) (2024)
- [SIMSHIFT: A Benchmark for Adapting Neural Surrogates to Distribution Shifts](https://mlsanity.com/paper/2506.12007/) (2025)
- [Size Lowerbounds for Deep Operator Networks](https://mlsanity.com/paper/2308.06338/) (2023)
- [Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operators](https://mlsanity.com/paper/2408.09004/) (2024)
- [Multi-scale DeepOnet (Mscale-DeepOnet) for Mitigating Spectral Bias in Learning High Frequency Operators of Oscillatory Functions](https://mlsanity.com/paper/2504.10932/) (2025)
- [Out-of-distribution generalization of deep-learning surrogates for 2D PDE-generated dynamics in the small-data regime](https://mlsanity.com/paper/2601.08404/) (2026)

## Built on by (most cited first, 6 of 6)

- [Semigroup Consistency as a Diagnostic for Learned Physics Simulators](https://mlsanity.com/paper/2605.26324/) (2026)
- [Operator Boosting Produces Pareto-Efficient PDE Surrogates](https://mlsanity.com/paper/2606.17460/) (2026)
- [Cellular Sheaf Neural Operators for Structure-Preserving Surrogate Modeling of Constrained PDEs](https://mlsanity.com/paper/2606.00937/) (2026)
- [Forecasting Without Full Identification: Reliability and Computational Cost in Clot-Growth Simulation](https://mlsanity.com/paper/2608.08165/) (2026)
- [Into the danger zone: stable extrapolation in high-dimensional function and operator learning](https://mlsanity.com/paper/2609.36709/) (2026)
- [Does Text Steer Neural PDE Surrogates? A Controlled Diagnostic with OperatorCLIP](https://mlsanity.com/paper/2609.38517/) (2026)

## Abstract (arXiv)

Neural PDE solvers are increasingly used as learned surrogates for families of partial differential equations, where the key machine learning challenge is not only interpolation on a fixed benchmark distribution but generalization under structured shifts in coefficients, boundary conditions, discretization, and rollout horizon. Yet evaluation is still often dominated by in-distribution test error, making robustness difficult to assess. We introduce a standardized stress-testing framework for neural PDE solvers under deployment-relevant shift. We instantiate it on three representative architectures -- Fourier Neural Operators (FNOs), a DeepONet-style model, and convolutional neural operators (CNOs) -- across five qualitatively different PDE families: dispersive, elliptic, multi-scale fluid, financial, and chaotic systems. Across 750 trained models, we measure robustness using baseline-normalized degradation factors together with spectral and rollout diagnostics. The resulting comparisons reveal that strong in-distribution accuracy does not reliably predict robustness, and that failure patterns depend jointly on architecture and PDE family. Our results provide a clearer basis for evaluating robustness claims in neural PDE solvers and suggest that function-space generalization under structured shift should be treated as a first-class evaluation target.