Physics-Informed Neural Networks construct lattice Dirac operators satisfying the Ginsparg-Wilson relation, reproducing overlap fermions to high accuracy and discovering a Fujikawa-type generalized relation via algebraic search.
Generalized Ginsparg-Wilson relations
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A lattice formulation of the Atiyah-Patodi-Singer index is built using spectral flow of domain-wall Dirac operators generalized beyond product boundaries and proven to recover the continuum index for small enough lattice spacing.
Reflection symmetry on twisted Dirac operators on warped cylinders holds precisely when 2A is integer, yielding unitary equivalence of APS blocks and an RO(O(2))-valued or mod-two spectral-flow invariant.
Minimal-doubling lattice fermion Hamiltonians yield single-Weyl phases when supplemented by a species-splitting mass term, but one-parameter symmetry-preserving deformations introduce additional Weyl nodes above a critical value.
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Capturing the Atiyah-Patodi-Singer index from the lattice
A lattice formulation of the Atiyah-Patodi-Singer index is built using spectral flow of domain-wall Dirac operators generalized beyond product boundaries and proven to recover the continuum index for small enough lattice spacing.