An Adjoint Formulation of Energetic Particle Confinement

TL;DR

The paper develops an adjoint formulation for energetic particle confinement in axisymmetric tokamaks and demonstrates that a Physics-Informed Neural Network can solve the corresponding inhomogeneous adjoint drift-kinetic equation to predict the escape time across phase space. By coupling a PINN with a GPU-accelerated particle-based drift-kinetic solver (JONTA), the study shows qualitative and regional quantitative agreement in predicting confinement structures, while highlighting challenges posed by large time-scale separations between fast orbital motion and slow collisions. The work provides a pathway toward rapid, data-augmented surrogates for fast-ion transport, suitable for integration into optimization loops, and outlines concrete extensions to handle 3D geometry, slowing-down physics, and broader fast-ion metrics. Overall, the adjoint-PINN framework offers a promising, mesh-free approach to characterizing fast-ion confinement with potential impact on tokamak design and operation.

Abstract

An adjoint formulation of energetic particle confinement in axisymmetric geometry is derived and evaluated using a Physics-Informed Neural Network (PINN). The PINN estimates the escape time of energetic ions by solving an inhomogeneous adjoint of the drift kinetic equation with a Lorentz collision operator, yielding predictions of the escape time of fast ions in tokamak geometry due to direct ion orbit loss and collisional transport. This is the first time a PINN has been used to solve the drift kinetic equation in tokamak geometry, a challenging problem due to the large time scale separation present between the rapid transit time of energetic ions, and their slow collision time scale. It is shown that a careful and intentional design of a PINN is able to learn the escape time for the majority of the geometry considered, suggesting a path toward constructing a rapid surrogate for use in a broader optimization framework.

Figures (10)

• Figure 1: Change of toroidal canonical momentum for 20 keV ions (panel a.) and 50 keV ions (panel b). Ten million deuterium ions were initialized randomly across the spatial and pitch domains. The time step was varied from $\Delta t = 0.2$ (blue curve), to $\Delta t = 0.1$ (orange curve), and $\Delta t = 0.05$ (green curve). The tokamak was assumed to have a minor radius of $a=0.5\;[\text{m}]$, an inverse aspect ratio of $a/R_0 = 1/3$, a magnetic field of $B_0 = 2\;[\text{T}]$, and a constant safety factor $q=2$.
• Figure 2: Circular flux surface geometry used in this work and example collisionless ion orbits. Panel (a) shows co-current ions with a passing ion with initial pitch $\xi=0.8$ (red curve), and a trapped ion with initial pitch $\xi=0.3$ (green curve). Panel (b) shows counter-current ions with a passing ion with initial pitch $\xi=-0.8$ (red curve), and a trapped ion with initial pitch $\xi=-0.3$ (green curve). Flux surface contours are shown in blue, with the example orbits shown in red. The black marker indicates the initial position of the ions, and arrows indicate their direction. The grad-B drift points downward in our example geometry. A deuterium ion with $20\;\text{keV}$, a tokamak with a minor radius of $0.5\;\text{m}$, inverse aspect ratio $a/R_0=1/3$, constant $q$-profile of $q=2$, and an on-axis magnetic field strength of $B_0=2\;\text{T}$ was assumed.
• Figure 3: Loss histories for 20 keV (panel a.) and 50 keV (panel b.) ions. Solid lines indicate the training loss whereas 'x' markers indicate the test loss. Six million training points were used and two million test points.
• Figure 4: (a) Escape time for an ion initialized in the counter-current direction with a pitch of $\xi = -0.8$, after different stages of training. The profile of the ion's escape time is shown in the panels (a), (b) and (c), with the residual indicated in panels (d), (e) and (f). The first column is after the SOAP phase of training (50,000 epochs), the second column is 10,000 into the SSBroyden phase (60,000 net epochs), and the final column is after 50,000 epochs of SSBroyden (100,000 epochs total). The quantity plotted is $\text{log}_{10}\left( 1+ T\right)$, yielding a $\text{log}_{10}$ scale for large values of $T$, but vanishing when $T=0$. A deuterium ion with $10\;\text{keV}$ was assumed.
• Figure 5: Test loss distribution for 20 keV ions. A uniform random distribution was used, where the size of the markers is proportional to the magnitude of the residual.