Data-Driven Scaling Laws for Confinement

Regression across many devices yields empirical scaling laws that predict energy-confinement time from engineering parameters, guiding new-device design.

What a scaling law is

An empirical scaling law expresses a plasma performance metric, most famously the energy-confinement time, as a power-law function of engineering parameters such as current, field, size, heating power, and density. Fitted across a multi-machine database, it lets designers extrapolate to devices not yet built.

The classic form

Confinement scalings are usually written as products of powers, for example a confinement time proportional to plasma current and size raised to positive powers and to heating power raised to a negative power. Taking logarithms turns the power law into a linear regression, which is fit by least squares over the database.

# log-linear fit of a confinement scaling, schematic
import numpy as np
X = np.log(np.column_stack([Ip, B, n, P, R]))  # engineering params
y = np.log(tau_E)
coef, *_ = np.linalg.lstsq(np.column_stack([np.ones(len(y)), X]), y, rcond=None)
# coef gives the exponents of the power law

Machine learning extensions

• Nonparametric fits (Gaussian processes, trees) that relax the power-law form
• Uncertainty on the prediction, crucial when extrapolating
• Detection of regime changes where a single power law fails
• Careful handling of correlated parameters across the database

Extrapolation risk

The danger is extrapolating far beyond the database. Engineering parameters are correlated in existing machines, so a fit can confound their individual effects, and a new device operating outside the data may not follow the law. Dimensionless scalings and physics-based constraints reduce, but do not eliminate, this risk.

Scaling laws inform target parameters for new concepts, but for a first-of-a-kind spherical-tokamak breeder like Hyperion, operating in a less-populated region of parameter space, they are used with explicit uncertainty and backed by first-principles modeling. Predicted confinement is a computational estimate for an unbuilt machine, not a guarantee.