Certificates for Complex-Compatible Learned Cochain Laplacians

Learning mesh-based operators from data can match training objectives while implicitly violating algebraic consistency constraints that classical discretizations satisfy by construction. Such violations can introduce near-kernel directions, degrade conditioning as resolution increases, and distort the low-frequency spectral structure on which downstream solvers and diagnostics rely. This work introduces a low-overhead compatibility certificate for learned operator pairs, together with a closed-form projection that maps a learned pair to its Frobenius-nearest chain-compatible operator. The certificate provides an explicit distance-to-compatibility and yields perturbation bounds for the discrete operator. These bounds imply stability guarantees for elliptic solves and for low-frequency spectral counts, provided a spectral gap separates the kernel from the rest of the spectrum and boundary treatments are well posed. Experiments on standard elliptic problems show that defect-aware training prevents condition-number blow-up at higher resolutions, improves robustness under mesh and topological distribution shifts, and maintains predictive accuracy relative to unconstrained learning. Overall, these results support the use of deployment-neutral, computable algebraic consistency checks to detect and control failure modes that are not revealed by loss values alone.

Many scientific machine-learning systems learn how to run simulations on meshes: simplified grids used to model physical spaces, materials, or fields. These systems can make accurate-looking predictions during training while quietly breaking basic consistency rules that reliable numerical methods normally satisfy. When this happens, the learned simulation may become unstable on finer grids, behave poorly on new mesh shapes, or give misleading information about the structure of the underlying domain. This paper introduces a practical test for detecting this hidden problem. The test measures how far a learned simulation operator is from satisfying the required consistency rule, and it also provides the closest consistent version for comparison. This makes it possible to predict when the learned operator is likely to remain stable, when its numerical solves are trustworthy, and when its low-frequency signals can still be interpreted reliably. Experiments show that controlling this consistency error prevents severe conditioning failures at higher resolutions and improves robustness when the mesh or topology changes, while maintaining strong predictive accuracy.