# Hadamard Flux Loss for Neural Free Boundaries

- ID: 1059
- Canonical URL: https://synthcore.org/idea/1059/hadamard-flux-loss-for-neural-free-boundaries
- API JSON: https://synthcore.org/api/idea/1059.json
- API Markdown: https://synthcore.org/api/idea/1059.md
- Verification status: unverified
- Source: [arXiv:2607.17187](https://arxiv.org/abs/2607.17187)
- Category: geometry
- Solves: accuracy, stability
- ML areas: loss, geometry, world-model
- Math tags: pde, measure-theory, differential-geometry, geometry
- Ratings: usefulness 5/10; difficulty 6/10; novelty 7/10

## Idea description

Use the paper's boundary Hadamard formula as a sensitivity-weighted interface objective for a neural potential and a neural implicit domain. Boundary points with large outward normal flux receive larger shape-update weight, while the positive mixed Monge–Ampère boundary measure supplies a geometry-aware quadrature weight. This gives a mathematically motivated alternative to uniformly weighted boundary residuals in neural free-boundary and obstacle-problem solvers.

## Mathematical statement

For a compact Kähler manifold with background form \(\omega_0\), let \(u_0\) be the equilibrium envelope, \(\omega_{u_0}=\omega_0+dd^c u_0\), and let \(U_\varepsilon=\{\rho<\varepsilon\}\) be an outward parallel deformation of a domain whose signed-distance function \(\rho\) is negative inside. The paper proves the one-sided Hadamard formula \(\left.\frac{d}{d\varepsilon}E_{\omega_0}(u_\varepsilon)\right|_{\varepsilon=0^+}=\frac{1}{n+1}\int_{\partial U}(-\partial_{\nu_{\rm out}}u_0)^2\,d\sigma_\rho^{\rm tot}\), where \(E_{\omega_0}\) is normalized Monge–Ampère energy, \(\nu_{\rm out}\) is the outward unit normal, \(\partial_{\nu_{\rm out}}u_0=\nabla u_0\cdot\nu_{\rm out}\), and \(d\sigma_\rho^{\rm tot}\) is a positive boundary reference measure. The associated divergence-measure flux current is \(\mathcal{J}_{\rm tot}=\frac{1}{V}d^c u_0\wedge\sum_{p=0}^{n-1}(p+1)\omega_{u_0}^{p}\wedge\omega_0^{n-1-p}\), with \(V=\int_X\omega_0^n\); its negative outward Anzellotti trace represents the interface Monge–Ampère mass. For a general normal velocity \(v_n\), adapt the parallel formula to \(\dot E\approx\frac{1}{n+1}\int_{\partial U}v_n q^2d\sigma_\rho^{\rm tot}\), where \(q=-\partial_{\nu_{\rm out}}u\ge 0\).

## Key formulas

- $$\left.\frac{d}{d\varepsilon}E_{\omega_{0}}(u_{\varepsilon})\right|_{\varepsilon=0^{+}}=\frac{1}{n+1}\int_{\partial U}\left(-\partial_{\nu_{\rm out}}u_{0}\right)^{2}d\sigma_{\rho}^{\rm tot}.$$
- $$\mathcal{J}_{\mathrm{tot}}=\frac{1}{V}d^{c}u_{0}\wedge\sum_{p=0}^{n-1}(p+1)\omega_{u_{0}}^{p}\wedge\omega_{0}^{n-1-p},\qquad \omega_{u_{0}}=\omega_{0}+dd^{c}u_{0},\qquad V=\int_X\omega_0^n.$$
- $$\dot E[v_n]\;\approx\;\frac{1}{n+1}\int_{\partial U}v_n(x)\,q(x)^2\,d\sigma_\rho^{\rm tot}(x),\qquad q(x)=-\nabla u_\theta(x)\cdot\nu_{\rm out}(x).$$
- $$\mathcal{L}_{\rm Had}=\frac{1}{n+1}\sum_{i=1}^{N_b}w_i\,q_i^2\,\max(0,v_{n,i})+\lambda_q\sum_{i=1}^{N_b}w_i\,[q_i-q_i^{\rm target}]^2.$$

## Implementation notes

(1) Integration point: represent the unknown interface by an MLP signed-distance field \(\rho_\phi(x)\), with \(U_\phi=\{x:\rho_\phi(x)<0\}\), and represent the potential by an MLP \(u_\theta(x)\). At every training step sample boundary points \(x_i\) by projecting random points to \(\rho_\phi=0\), or by using a narrow-band sampler. Compute \(\nu_i=\nabla\rho_\phi(x_i)/(\|\nabla\rho_\phi(x_i)\|+\epsilon)\), the outward normal flux \(q_i=-\nabla u_\theta(x_i)\cdot\nu_i\), and the proposed boundary loss or shape-gradient signal. Add this to the PDE residual, obstacle/contact loss, and signed-distance eikonal loss. (2) Pseudocode: sample \(x_i\); project to \(\rho_\phi(x_i)=0\); compute \(\nu_i\), \(q_i\), and positive weights \(w_i\); estimate normal interface motion \(v_{n,i}\) from the shape-network update or prescribe a small outward probe \(v_n=1\); set \(L_{\rm Had}=\frac{1}{n+1}\sum_iw_iq_i^2\max(0,v_{n,i})\); backpropagate \(L=L_{\rm PDE}+\lambda_HL_{\rm Had}+\lambda_{\rm eik}L_{\rm eik}\). Stop gradients through \(w_i\) during the first experiment to isolate the flux effect. (3) Mathematical quantities versus estimates: \(q_i\) is obtained by automatic differentiation; \(\nu_i\) is obtained from the implicit surface; \(w_i\) approximates \(d\sigma_\rho^{\rm tot}\). In complex dimension one, use the scalar density proxy \(w_i\propto |\nabla u_\theta(x_i)|\) or the absolute normal trace of \(d^cu\); in higher dimensions estimate the mixed Monge–Ampère density from determinants of \(\omega_\theta=\omega_0+dd^cu_\theta\), using the paper's coefficients \((p+1)\) in the current. Normalize weights to have mean one. Validate the formula separately by evaluating the neural energy at \(\rho_\phi\) and at \(\rho_\phi+\varepsilon\), checking that the finite difference matches \((n+1)^{-1}\sum_iw_iq_i^2\). (4) First experiment: use a two-dimensional real analogue of the obstacle/free-boundary problem on a unit square or disk, with a known radial solution and an MLP potential plus circle-like implicit boundary. Compare uniform boundary residual weighting against the Hadamard weighting at equal network size and optimization steps. Measure interface Hausdorff error, PDE residual, contact-condition violation, and agreement between predicted and finite-difference energy derivatives. A successful signal is lower interface error and fewer unstable boundary oscillations, together with at least 20% lower derivative-prediction error or faster reduction of the free-boundary residual.

## Disclaimer

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