Improving ideal MHD equilibrium accuracy with physics-informed neural networks

Abstract

We present a novel approach to compute three-dimensional Magnetohydrodynamic equilibria by parametrizing Fourier modes with artificial neural networks and compare it to equilibria computed by conventional solvers. The full nonlinear global force residual across the volume in real space is then minimized with first order optimizers. Already,we observe competitive computational cost to arrive at the same minimum residuals computed by existing codes. With increased computational cost,lower minima of the residual are achieved by the neural networks,establishing a new lower bound for the force residual. We use minimally complex neural networks,and we expect significant improvements for solving not only single equilibria with neural networks,but also for computing neural network models valid over continuous distributions of equilibria.

Figures (7)

• Figure 1: Illustration of the fully differentiable computational graph [id=TT]of our contribution with parameters $\pmb{\nu}_X$ and associated derivatives $\partial_{\pmb{\nu}_X} ||\mathbf{F}||_2$. The inverse Fourier transform is marked with $\mathcal{F}^{-1}$. [id=TT]The strong form of the ideal residual requires up to second order gradients.Computing the force residual requires first and second order gradients of $(R,\lambda,Z)$ with respect to $\rho$.
• Figure 2: Poincare sections for a D-shaped tokamak test-case with boundary defined by \ref{['eqn:dshape_def']}. Dashed green lines represent the solution with $n_\mathrm{l}=8$ hidden nodes and full red lines represent the VMEC solution,revealing a qualitative agreement in flux surface geometry. Profiles of $\theta^{\star}=\theta + \lambda(\rho,\theta,\zeta)$ are plotted at eight $\theta^{\star}$ locations from axis to boundary.
• Figure 3: Normalized flux surface averaged force error for the solutions of VMEC, and the DESC for the D-shaped tokamak test-case plotted over $\rho$. The optimization continued for about 100 times longer than DESC and VMEC but achieves lowest minimum force error over the whole volume. VMEC's force error spikes at $\rho<0.2$.
• Figure 4: Poincare section for a $M_\mathrm{b}=N_\mathrm{b}=12$ equilibrium of the stellarator solved with with $n_\mathrm{l}=18$ hidden nodes in dashed green and VMEC in red. Both solutions agree with each other qualitatively. $\theta^{\star}=\theta + \lambda(\rho,\theta,\zeta)$ profiles are plotted at eight $\theta^{\star}$ locations from axis to boundary.
• Figure 5: Normalized flux surface averaged force error for VMEC, and DESC for the $M_\mathrm{b}=N_\mathrm{b}=12$ test-case plotted over $\rho$. The achieve lowest minimum force error over the whole volume and the force residual of VMEC spikes at $\rho<0.15$.