Classical optimisation and the NF2 loss

NF2’s training loss is closely related to the classical optimisation functional for NLFFF extrapolation.

That is the important point: NF2 is not solving different physics. It is minimising the same force-free and divergence residuals, but with a neural-field representation instead of grid values.

Same physics functional

The classical optimisation method minimises a volume integral:

$${\mathcal{L}}_{\text{classical}}={\int }_V[w_f\frac{|(\nabla \times \mathbf{B})\times \mathbf{B}{|}^2}{|\mathbf{B}{|}^2}+w_d|\nabla \cdot \mathbf{B}{|}^2]dV.$$

NF2 minimises sampled versions of the same two physics terms, plus a boundary term:

$${\mathcal{L}}_{\text{NF2}}={\lambda }_{\text{ff}}⟨\frac{|(\nabla \times \mathbf{B})\times \mathbf{B}{|}^2}{|\mathbf{B}{|}^2}⟩+{\lambda }_{\text{div}}⟨|\nabla \cdot \mathbf{B}{|}^2⟩+{\lambda }_{\text{b}}⟨|\mathbf{B}-{\mathbf{B}}_{\text{obs}}{|}^2⟩.$$

The two physics terms are the same. The differences are mostly numerical bookkeeping:

• the classical volume integral becomes an average over sampled collocation points;
• the weighting functions $w_f,w_d$ become scalar weights and spatial loss scalers;
• the classical fixed boundary becomes a soft NF2 boundary penalty.

That last point is a genuine modelling difference. NF2 can trade boundary agreement against physics residuals.

Different function space

The real difference is the space being searched.

	Classical optimisation	NF2 / PINN
Field representation	grid values	continuous network $\mathbf{B}={\mathcal{N}}_{\theta }(\mathbf{x})$
Unknowns	field components on the grid	network weights $\theta$
Derivatives	finite differences	autograd
Integral estimate	quadrature over cells	sampled collocation points
Optimiser	artificial relaxation	Adam / SGD-style optimisation
Boundary	usually fixed	weighted loss term

In the classical method, the unknowns are the field values themselves. In NF2, the unknowns are the weights of a SIREN.

The network acts as an implicit regulariser. It cannot represent every grid-scale pattern freely, so it tends to produce smoother fields. That can help with noisy data, but it can also smooth away real small-scale structure.

Why this matters for the dissertation

This parallel is useful for three reasons.

First, it sets a fair bar. NF2 is a different numerical route through the same inverse problem, not a new physical model.

Second, it explains failure modes. Because the boundary is a variable loss term, NF2 can reduce total loss by giving up some boundary agreement. A classical fixed-boundary method cannot make that exact trade.

Third, it frames the contribution. Any NF2 advantage is an advantage of representation and optimisation: smooth neural fields, analytic derivatives, mesh-free evaluation, and flexible boundaries. Any disadvantage, such as hyperparameter sensitivity or soft-boundary drift, is the cost of that choice.

Related

• Classical NLFFF extrapolation methods
• NF2 training loop and loss scheduling
• Potential-field loss scaling in NF2
• NLFFF quality metrics
• Residuals in NF2 and PINNs