PINNs for Inverse Problems

By treating unknown coefficients as trainable parameters, a PINN reconstructs the breeder's hidden profiles and the burner's potential directly from diagnostics.

Forward becomes inverse

The same PINN that solves a PDE forward can identify unknown terms in it. The unknown profiles - p'(psi) and F F'(psi) for the breeder, or the ambipolar potential parameters for the burner - are made trainable alongside the network weights, and the loss adds a data term matching diagnostics. Minimizing residual and data mismatch jointly recovers both the field and the hidden coefficients.

Inverse PINN objective:

  minimize over (theta, lambda):
    L_pde(theta, lambda) + w_data * L_data(theta)

  theta  : network weights (the field psi)
  lambda : unknown physical coefficients (profiles, potential)

  PDE residual couples lambda to the field; data pins the field.

Regularization is essential

Inverse problems are ill-posed: many profile choices fit the data. The PINN adds smoothness and physical-admissibility priors on lambda - monotonicity, sign, bounded curvature - so the recovered profiles are physical, not noise-fit. This mirrors the regularization in classical equilibrium reconstruction but is embedded in one differentiable objective.

# joint field + profile identification (schematic)
theta = init_net(); lam = init_profiles()
for it in range(max_it):
    L = pde_residual(theta, lam) \
      + w_d * data_mismatch(theta, diagnostics) \
      + w_r * smoothness(lam) + w_s * sign_penalty(lam)
    theta, lam = adam_step(L, [theta, lam])   # both trainable

Application on both machines

For the breeder this yields real-time equilibrium reconstruction that is a physically admissible Grad-Shafranov solution by construction, feeding the twin. For the burner it recovers the ambipolar potential profile consistent with quasineutrality from end-cell diagnostics - but with the standing caveat that the operating regime is 166-830x beyond any device, so the inversion is extrapolative and its uncertainty is reported, not suppressed.

• Breeder: recover p'(psi), F F'(psi), and psi from magnetics + kinetics.
• Burner: recover Phi(z) consistent with quasineutrality and ambipolarity.
• Output: field plus profiles plus a reported uncertainty band.

Inverse PINNs share the twin's honesty rule: where diagnostics are sparse or the regime is extrapolative, the recovered coefficients carry wide bands and downstream control widens its margins.