Hard-Constraint PINNs for Boundary Conditions

Kronos bakes boundary conditions into the network architecture so they hold exactly, improving conditioning over soft-penalty PINNs.

Soft vs hard constraints

A standard PINN enforces boundary conditions with a penalty term, so they hold only approximately and compete with the interior residual during optimization. For equilibrium problems where the boundary flux value anchors the whole solution, an imperfect boundary is a real error source. Kronos uses hard-constraint parameterizations that satisfy the boundary by construction.

# hard-constrained ansatz: BC satisfied exactly for any network N
#   psi(x) = g(x) + D(x) * N_theta(x)
#   g(x) : a function meeting the boundary data (psi = psi_bnd on dOmega)
#   D(x) : a distance-like function, D = 0 exactly on dOmega
# => on the boundary psi = g = psi_bnd, regardless of N_theta

With this ansatz the boundary term disappears from the loss, leaving only the physics residual (and any data term). The optimization is better conditioned because it no longer trades boundary accuracy against interior accuracy, and training converges faster, which matters when Kronos retrains PINNs in batch as the design evolves pre-FOAK.

Where Kronos applies it

• Breeder equilibrium: fixed edge flux on the reconstruction boundary
• Burner potential: known potential at the DEC collector and central-cell reference
• Divertor thermomechanics: prescribed surface temperature/flux boundaries
• Symmetry: axisymmetry (breeder) and mirror symmetry (burner) as exact constraints

Enforcing symmetry as a hard constraint is especially valuable: the breeder is axisymmetric and the burner mirror is symmetric about its midplane, so encoding these directly halves the effective domain and removes a class of spurious asymmetric solutions the network could otherwise drift into.

Hard constraints do not remove the need for validation; they remove one error mode. The interior physics residual is still checked against reference solvers, and the network's output is still confidence-scored before the twin uses it.