Collocation Sampling and Adaptive Refinement

Where a PINN enforces its physics residual determines its accuracy; Kronos samples collocation points adaptively around steep gradients like the plasma edge.

The role of collocation points

A PINN enforces the PDE only where it evaluates the residual, on a set of collocation points. Uniform sampling wastes capacity in smooth regions and under-resolves the places that matter: the steep pressure gradient at the negative-triangularity edge, the X-point singularity, and the burner's plug-throat transition where the field rises to 26.49 T. Kronos uses residual-adaptive sampling to concentrate points where the physics residual is largest.

Adaptive refinement loop

# residual-based adaptive collocation (RAR-style)
pts = sample_uniform(domain, N0)
for round in range(R):
    train_pinn(pts)
    r = abs(pde_residual(pinn, dense_candidates))
    hot = topk(dense_candidates, key=r, k=M)   # high-residual regions
    pts = concat(pts, hot)                       # add where physics is hard

The refinement targets exactly the features control cares about. For the breeder, the pedestal and separatrix get dense sampling so the reconstructed boundary and the peeling-ballooning margin are accurate. For the burner, the mirror throat and the ambipolar barrier get refined so the plug-density and potential solve resolves the sharp axial gradients.

Cost control

Collocation cost is an offline (L0) concern: adaptive refinement happens during batch training on the GPU clusters, producing a trained network whose inference cost is fixed and mesh-free. The runtime twin never re-samples; it evaluates the already-trained PINN. This split, expensive adaptive training offline, cheap deterministic inference online, is what lets a physically faithful solver live inside the 50-100 ms shadow budget.

Sampling quality is part of validation: a PINN whose residual is low only because it was tested on its own training points is rejected. Kronos evaluates residuals on held-out dense grids and against reference finite-element solutions before a PINN is allowed into the twin.