Dynamical phase retrieval for Schr{ö}dinger evolution on finite graphs
arXiv:2607.26705
2026
Architecture
2 ideas extracted · analyzed Aug 31, 2026
What the math gives to ML
The paper gives a constructive observability principle for recovering a complex state from coordinatewise magnitudes observed along a known real symmetric dynamical system. Its transferable asset is not the graph Laplacian itself, but the spectral design criterion: nonresonant eigenvalue combinations, an invertible squared-eigenvector matrix, and overlap of eigenvector supports make magnitude-only trajectories injective up to a global phase. This can be transplanted into complex recurrent or message-passing networks as a fixed, phase-preserving phaseless sensing layer, or used to regularize a learned generator so that its hidden states remain identifiable from intensity measurements.
Ideas from this paper
△ Mechanism confirmed, baseline not beaten
2026
Build a complex-valued recurrent or graph-neural layer whose hidden state evolves under a fixed graph Schrödinger operator and is exposed to the downstream network only through coordinate magnitudes at several times. Choose the diagonal potential so that the spectrum has unique unordered pair sums, the squared-eigenvector matrix is invertible, and every eigenvector pair overlaps in at least one observed coordinate; the resulting temporal intensity code is theoretically injective up to one…
Useful7/10
Difficulty6/10
Novelty8/10
Unverified
2026
When the Schrödinger generator is learned, regularize its spectrum and eigenvectors so that the magnitude trajectory remains well-conditioned for recovering hidden complex states. Penalize small singular values of the squared-eigenvector matrix and near-colliding eigenvalue pair sums, preventing a learned dynamical layer from becoming spectrally invisible or phase-ambiguous.
Useful6/10
Difficulty6/10
Novelty7/10