# Full-Field Phase-Boundary PINN

- ID: 2288
- Canonical URL: https://synthcore.org/idea/2288/full-field-phase-boundary-pinn
- API JSON: https://synthcore.org/api/idea/2288.json
- API Markdown: https://synthcore.org/api/idea/2288.md
- Verification status: unverified
- Source: [arXiv:2608.14348](https://arxiv.org/abs/2608.14348)
- Category: regularization
- Solves: stability, accuracy, generalization
- ML areas: mlp, training, regularization
- Math tags: pde, harmonic-analysis, symmetry
- Ratings: usefulness 6/10; difficulty 5/10; novelty 9/10

## Idea description

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.

## Mathematical statement

The full-field formulation solves E''(x)+k^2E(x)=0 on 0<=x<=a with phase multiplier z=exp(iqa) and boundary equations E(a)=zE(0), E'(a)=zE'(0). In contrast, the periodic-factor formulation uses E=\widetilde{E}e^{iqx} and the volume operator L(q)\widetilde{E}=\widetilde{E}''+2iq\widetilde{E}'+(k^2-q^2)\widetilde{E}. The proposed PINN uses the full-field residual and boundary loss. Here E_    heta is the neural field, k is the eigenvalue parameter, q is the prescribed Bloch wavenumber, z=e^{iqa}, and derivatives are obtained by automatic differentiation. Reciprocal shifts q'=q+m\kappa leave z unchanged because e^{i(q+m\kappa)a}=e^{iqa}e^{i2\pi m}=e^{iqa}, so the full-field boundary problem is invariant under q -> q+m\kappa. Eigenvectors remain ambiguous up to a nonzero complex scalar, so normalization and phase fixing are required.

## Key formulas

- $$E^{\prime\prime}(x)+k^2E(x)=0,\qquad E(a)=zE(0),\quad E'(a)=zE'(0),\quad z=e^{iqa},$$
- $$\mathcal{L}_{\mathrm{FF}}(E_\theta,k)=\frac{1}{N}\sum_{j=1}^{N}\left|E_\theta''(x_j)+k^2E_\theta(x_j)\right|^2,$$
- $$\mathcal{L}_{\mathrm{bc}}=|E_\theta(a)-zE_\theta(0)|^2+|E_\theta'(a)-zE_\theta'(0)|^2,$$
- $$e^{i(q+m\kappa)a}=e^{iqa},\qquad \kappa=\frac{2\pi}{a},\quad m\in\mathbb{Z}.$$

## Implementation notes

Use a complex-valued MLP E_theta(x), implemented as an MLP with two real outputs for real and imaginary parts, as a physics-informed eigenmode solver. The exact integration point is the training loss. Sample interior points x_j in [0,a], evaluate E_theta and its first and second derivatives by automatic differentiation, and optimize both network weights and a scalar eigenvalue parameter k or k^2. Compute z=cos(qa)+i sin(qa), r_j=E_theta''(x_j)+k^2E_theta(x_j), r_a=E_theta(a)-zE_theta(0), and d_a=E_theta'(a)-zE_theta'(0). Minimize L=mean_j|r_j|^2+lambda_bc(|r_a|^2+|d_a|^2)+lambda_norm(|E_theta(x_ref)|^2-1)^2. To remove arbitrary global phase, either penalize Im(E_theta(x_ref))^2 or align the prediction to a reference field during evaluation. Do not include the periodic-factor residual with 2iq and q^2 terms in the first experiment. The residual and phase conditions come directly from the mathematics; collocation density, loss weights, and initialization are empirical. Begin with the homogeneous cell, where exact solutions are E=exp(+-ikx), then test a piecewise-constant periodic coefficient and compare against a finite-element generalized eigenproblem. Sweep q and q+m\kappa and verify that recovered eigenvalues and reconstructed fields coincide. Success is fewer low-residual nonphysical modes, stable convergence across aliases, and lower eigenvalue error than a periodic-factor PINN at equal parameter count.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.