When PDE Solvers Break: Unmasking the Limits of Neural Operators

A new study systematically tests the robustness of Fourier Neural Operators across a range of partial differential equations, revealing common failure points and highlighting areas for improvement.
![Market volatility, measured by a six-factor norm, consistently spikes during periods of documented stress-clustering into a distinct “crisis” regime [latex]\text{(red)}[/latex]-while moderate fluctuations define an “elevated” regime [latex]\text{(yellow)}[/latex], suggesting that fear and response to major events predictably shape market behavior over the 1990-2024 period.](https://arxiv.org/html/2601.10732v1/x1.png)






![The study demonstrates how the dynamics of neuronal firing rates-specifically, the mean firing rate and membrane potential-shift across regimes determined by the correlation of noise, with theoretical predictions aligning with simulations of quadratic integrate-and-fire neurons, and revealing a dependence on both the correlation coefficient and noise intensity as described by the deterministic component of [latex]Eq. (9)[/latex].](https://arxiv.org/html/2601.10032v1/x2.png)