Variational principles for the interaction of liquid crystals and electric fields in the Oseen--Frank model
arXiv:2607.23315
2026
Optimization
2 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper converts an electrostatic liquid-crystal problem from a difficult saddle-point formulation into nested minimizations using convex duality and Hodge decomposition. The transferable asset is the elimination of an inner scalar PDE saddle variable: a divergence-constrained flux represented by a curl has an automatically satisfied conservation law and a positive quadratic objective. This can stabilize neural PDE solvers and differentiable energy models by replacing jointly optimized primal and dual fields with a convex inner minimization. The paper also provides a quadratic small-anisotropy energy error estimate that can justify approximate electrostatic losses and infrequent exact field solves.
Ideas from this paper
✓✓ Beats tuned baseline
2026
Replace a jointly optimized scalar electrostatic potential in a neural PDE solver with a dual flux represented by a Hodge curl correction. The resulting inner problem is a positive quadratic minimization with the divergence constraint satisfied exactly, avoiding unstable primal-dual training dynamics.
Useful7/10
Difficulty6/10
Novelty8/10
Unverified
2026
Use the small dielectric-anisotropy estimate to replace expensive nonlocal electrostatic solves with a local field-energy surrogate during most neural-network updates. Periodically evaluate the exact field, estimate the approximation constant, and trigger correction solves only when the observed error exceeds the predicted quadratic scale.
Useful6/10
Difficulty4/10
Novelty7/10