Full-field and Bloch-periodic-factor discretizations: Accuracy and phantom modes
arXiv:2608.14348
2026
Architecture
2 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper identifies a concrete failure mode in periodic discretizations: representing a Bloch mode as a periodic factor with wavenumber-dependent differential terms can break reciprocal-lattice covariance and create nonphysical phantom modes, even when the continuous full-field and periodic-factor formulations are equivalent. The transferable asset is the exact gauge symmetry relating equivalent parameterizations under q -> q + m kappa. Neural operators, Fourier-feature models, and Bloch-conditioned surrogates should either canonicalize the Bloch parameter into one Brillouin zone or explicitly enforce the corresponding transformation of the periodic factor. A second route is to impose the full-field boundary-phase formulation in physics-informed eigenmode networks, avoiding unconstrained q-dependent volume operators.
Ideas from this paper
✓✓ Beats tuned baseline
2026
Build a Bloch-conditioned neural model whose periodic-factor representation transforms covariantly when the supplied Bloch wavenumber is shifted by a reciprocal lattice vector. Either canonicalize q to the first Brillouin zone or augment training with mathematically paired examples whose outputs differ by the exact phase gauge. This prevents the network from learning inconsistent predictions for physically identical Bloch modes.
Useful7/10
Difficulty4/10
Novelty8/10
Unverified
2026
For neural eigenmode solvers on periodic domains, train the full field directly and impose Bloch phase coupling only at opposite cell boundaries, rather than differentiating a periodic factor with respect to q through a quadratic volume operator. The boundary formulation preserves reciprocal-lattice equivalence exactly through z=exp(iqa), reducing spurious eigenmodes caused by inconsistent q-dependent discretization.
Useful6/10
Difficulty5/10
Novelty9/10