Tensor network approaches for plasma dynamics

TL;DR

The paper investigates applying tensor-network methods to plasma dynamics governed by the Vlasov–Maxwell system and ideal MHD, aiming to overcome the curse of dimensionality in high-dimensional phase-space simulations. It develops Matrix Product State encodings for distribution functions and fields, analyzes bond-dimension growth under external magnetic fields, and demonstrates that comb tensor networks can achieve similar accuracy with substantially fewer resources than large MPS in beat-wave scenarios. The work includes validation against Particle-In-Cell codes and demonstrates MHD simulations with turbulence-like features such as the Orszag–Tang vortex, showing convergence with increasing bond dimension and stability when using a Shuman filter. Overall, the results indicate that TNs offer a viable, compressive framework for kinetic and fluid plasma models, with clear guidance on when to prefer MPS vs comb geometries and how to handle stability and accuracy.

Abstract

The dynamics of plasmas are governed by a set of non-linear differential equations which remain challenging to solve directly for large 2D and 3D problems. Here we investigate how tensor networks could be applied to plasmas described by the Vlasov-Maxwell system of equations and investigate parameter regimes which show promise for efficient simulations. We show for low-dimensional problems that the simplest form of tensor networks known as a Matrix Product State performs sufficiently well, however in regimes with a strong permanent magnetic field or high-dimensional problems one may need to consider alternative tensor network geometries. We conclude the study of the Vlasov-Maxwell system with the application of tensor networks to an industrially relevant test case and validate our results against state of the art plasma solvers based on Particle-In-Cell codes. We also extend the application of tensor networks to the alternative plasma description of Magnetohydrodynamics and outline how this can be encoded using Matrix Product States.

Figures (23)

• Figure 1: Three possible MPS decomposition for a 1D1V plasma system. We show two possible sequential orderings in $a)$, where the $x$ dimension is encoded from coarsest to finest scale followed by the velocity dimension in same order, and $b)$, where the $x$ dimension is encoded from finest to coarsest and the velocity from coarsest to finest. $c)$ shows an inter-leafed ordering where one alternates between an $x$ and $v$ tensor, but grouping together length scales of each dimension.
• Figure 2: Snapshots of a cut of the electron distribution $f_e(x,v_1)$ through phase space for the bump-on-tail instability. We show a slice through the $v_2=0$ plane, resulting in a 2D function in $x$ and $v_1$, at various simulation times of $a) t=7$, $b) t=14$, $c) t=18$ and $d) t=22$. These simulations where conducted on a $2^8 \times 2^8 \times 2^8$ phase space grid, resulting in a $24$ site MPS encoding, and evolved via a fixed time-step $4th$ order Runge kutta scheme with a time-step of $dt=0.0014$. A truncation cutoff of $10^{-15}$ was used to adaptively control the bond dimension $\chi$.
• Figure 3: Squared singular values of the MPS representation of the electron distribution function $f_e$ in descending order during the bump-on-tail simulation. We plot the singular values across the a)$n=7$ b)$n=8$ and c) $n=9$ bond of the MPS at simulation times of $t=7$ (red dots), $t=14$ (blue square), $t=18$ (green diamond) and $t=22$ (pink stars). At all times and at all bonds the singular values decay fast indicating the ability to effectively truncate the MPS representation.
• Figure 4: Results from simulation of Eq. \ref{['eq:mag_init']} on an $2^8\times2^8\times2^8$ phase space grid for various magnetic field strengths $B_o$. $a)$ Maximum of electric field and $b)$ maximal bond dimension of $K_1$ intermediate MPS as a function of simulation time $t$ for external magnetic field strengths of $B=0$ (solid blue), $B=0.01$ (dotted red), $B=0.05$ (dashed green) and $B=0.1$ (dash dotted black). $c)$ Maximal bond dimensions of $K_1$ MPS as a function of external magnetic field strength $B_o$ at simulation times of $t=3.5$ (blue dots), $t=6.1$ (red squares), $t=9.3$ (green diamonds) and $t=12.2$ (black triangles).
• Figure 5: Results from simulation of Eq. \ref{['eq:mag_init']} on an $2^8\times2^8\times2^8$ phase space grid for various magnetic field strengths $B_o$. We plot a slice through phase space along the $x=\frac{L}{2}$, $v_2=0$ plane of the change in the electron distribution relative to the initial condition $\delta f= f(t)-f(t=0)$ at a simulation time of $t=15$. We show this slice for the case of $B_0=0$ (solid black), $B_o=0.01$ (dotted red) and $B_o=0.1$ (dashed blue).