What neural operators really learn: beyond matching outputs

Neural operators trained to solve PDEs are evaluated almost entirely on prediction error, but this masks fundamental flaws: a model can match solutions while exhibiting wrong sensitivities or unstable dynamics. Researchers developed a Jacobian-based spectral audit that treats the network's output derivatives as a learned tangent operator, then projects it onto Fourier modes to reveal frequency response, phase structure, and mode coupling. Testing on benchmark problems, the audit caught failures invisible to prediction metrics—including high-frequency degradation and incorrect phase recovery—while also diagnosing operator inconsistencies even when pointwise predictions looked good.