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.
Use the paper's order-parameter dynamics to initialize spectral feature modes with deliberately separated activation times. This creates a controlled progressive-learning curriculum in which dominant modes become available first and weaker modes activate later, potentially reducing early gradient interference.