Application of parametric Shallow Recurrent Decoder Network to magnetohydrodynamic flows in liquid metal blankets of fusion reactors

M. Lo Verso; C. Introini; E. Cervi; L. Savoldi; J. N. Kutz; A. Cammi

cs.LGApr 2, 2026

Application of parametric Shallow Recurrent Decoder Network to magnetohydrodynamic flows in liquid metal blankets of fusion reactors

Authors: M. Lo Verso, C. Introini, E. Cervi, L. Savoldi, J. N. Kutz, A. Cammi

Organizations: Department of Energy, Politecnico di Milano, Milano, 20133, Italy · MATHEP Group, Dept. of Energy "Galileo Ferraris", Politecnico di Torino, Torino, Italy · Autodesk Research, 6 Agar Street, London UK · Department of Mechanical and Nuclear Engineering & Emirates Nuclear Technology Center, Khalifa University, Abu Dhabi, 127788, United Arab Emirates

Abstract

Magnetohydrodynamic (MHD) phenomena play a pivotal role in the design and operation of nuclear fusion systems, where electrically conducting fluids (such as liquid metals or molten salts employed in reactor blankets) interact with magnetic fields of varying intensity and orientation, influencing the resulting flow dynamics. The numerical solution of MHD models entails the resolution of highly nonlinear, multiphysics systems of equations, which can become computationally demanding, particularly in multi-query, parametric, or real-time contexts. This study investigates a fully data-driven framework for MHD state reconstruction that integrates dimensionality reduction through Singular Value Decomposition (SVD) with the SHallow REcurrent Decoder (SHRED), a neural network architecture designed to reconstruct the full spatio-temporal state from sparse time-series measurements of selected observables, including previously unseen parametric configurations. The SHRED methodology is applied to a three-dimensional geometry representative of a portion of a WCLL blanket cell, in which lead-lithium flows around a water-cooled tube. Multiple magnetic field configurations are examined, including constant toroidal fields, combined toroidal-poloidal fields, and time-dependent magnetic fields. Across all considered scenarios, SHRED achieves high reconstruction accuracy, robustness, and generalization to magnetic field intensities, orientations, and temporal evolutions not seen during training. Notably, in the presence of time-varying magnetic fields, the model accurately infers the temporal evolution of the magnetic field itself using temperature measurements alone. Overall, the findings identify SHRED as a computationally efficient, data-driven, and flexible approach for MHD state reconstruction, with significant potential for real-time monitoring, diagnostics and control in fusion reactor systems.

Figures & tables

Figure 1: Schematic representation of the SHRED architecture applied to the considered three-dimensional test case. After compressing the original dataset via SVD, three sensors record the local temporal evolution of the temperature field. The resulting time-series signals are encoded into a latent space through a Long Short-Term Memory (LSTM) network. Subsequently, a Shallow Decoder Network (SDN) maps the latent representation onto a compressed approximation of the full set of spatio-temporal field variables. Finally, the SVD basis is used to reconstruct the corresponding full-order state back-projcting its compressed representation.

Figure 2: Visual sketch of the selected model geometry. The image of the elementary cell on the left is taken from [ 40 ] .

$\rho_{0}$	$9806\ \text{kg/m}^{3}$	$\kappa$	$20.93\ \text{W m}^{-1}\ \text{K}^{-1}$	$p_{\text{ext}}$	$10^{5}\ \text{Pa}$
$\mu$	$1.93\times 10^{-3}\ \text{Pa}\cdot\text{s}$	$c$	$189.5\ \text{J kg}^{-1}\text{K}^{-1}$	$\mathcal{N}_{h}$	$112640$
$\mu_{B}$	$1.26\times 10^{-6}\ \text{H/m}$	$u_{0}$	$0.01\ \text{m/s}$	$L$	$0.07\ \text{m}$
$\sigma$	$7.82\times 10^{5}\ \Omega^{-1}\text{m}^{-1}$	$T_{0}$	$600\ \text{K}$	$a$	$0.02\ \text{m}$
$\beta$	$1.3\times 10^{-4}\ \text{K}^{-1}$	$T_{\text{pipe}}$	$560\ \text{K}$	$r$	$0.005\ \text{m}$

Table 1: Physical and numerical parameters for the FOM.

Figure 3: Overview of the selected magnetic field configurations considered in this analysis: a) constant toroidal magnetic field, b) constant toroidal toroidal + poloidal magnetic field, c) time-varying magnetic field.

Sensors	x [m]	y [m]	z [m]
S1	0.0070	0.0014	0.0617
S2	0.0051	-0.0018	0.0564
S3	-0.0027	0.0045	0.0066

Table 2: Positions of the measurement sensors in the domain.

Figure 4: Subdivision of the dataset in training, validation and test snapshots.

Figure 5: Results for the temperature (first column), velocity (second column) and pressure (third column) for the case with $B_{x}=0.75\,T$ at time $t=2\,s$ . The first row displays the reference full-order solution while the second row shows the reconstruction produced by the SHRED model. The third row reports the absolute difference between the FOM and the SHRED.

Figure 6: Results for the temperature (first column), velocity (second column) and pressure (third column) for the case with $B_{x}=1.85\,T$ at time $t=2\,s$ . The first row displays the reference full-order solution while the second row shows the reconstruction produced by the SHRED model. The third row reports the absolute difference between the FOM and the SHRED.

Figure 7: Results for the temperature (first column), velocity (second column) and pressure (third column) for the case with $B_{x}=2.5\,T$ at time $t=2\,s$ . The first row displays the reference full-order solution while the second row shows the reconstruction produced by the SHRED model. The third row reports the absolute difference between the FOM and the SHRED.

Figure 8: Temporal behavior of relative $L^{2}$ -error of the SHRED reconstruction over time for temperature, velocity and pressure for the cases with $B_{x}=0.75\ T$ (a), $B_{x}=1.85\ T$ (b) and $B_{x}=2.5\ T$ (c).

Snapshots	$B_{x}$	$B_{y}$	$\|\mathbf{B}_{tot}\|$	$\alpha$
Training	$1.0\,\text{T}$	$0.2\,\text{T}$	$1.02\,\text{T}$	$11.31^{\circ}$
	$1.4\,\text{T}$	$0.35\,\text{T}$	$1.44\,\text{T}$	$14.04^{\circ}$
	$1.8\,\text{T}$	$0.55\,\text{T}$	$1.88\,\text{T}$	$16.99^{\circ}$
	$2.0\,\text{T}$	$0.7\,\text{T}$	$2.12\,\text{T}$	$19.29^{\circ}$
Validation	$1.2\,\text{T}$	$0.3\,\text{T}$	$1.24\,\text{T}$	$14.04^{\circ}$
Test	$1.6\,\text{T}$	$0.45\,\text{T}$	$1.66\,\text{T}$	$15.71^{\circ}$

Table 3: Magnetic field configurations and subdivision of snapshots into training, validation and test sets.

Figure 9: Results for the temperature (first column), velocity (second column) and pressure (third column) for the case with $B_{x}=1.60\,\text{T}$ and $B_{y}=0.45\,\text{T}$ at time $t=2\,s$ . The first row displays the reference full-order solution while the second row shows the reconstruction produced by the SHRED model. The third row reports the absolute difference between the FOM and the SHRED.

Figure 10: Temporal behavior of relative $L^{2}$ -error of the SHRED reconstruction over time for temperature, velocity and pressure for the test case with $B_{x}=1.6\ T$ and $B_{y}=0.45\,\text{T}$ .

Set	$A\,[\text{T}]$	$\omega\,[1/\text{s}]$	$\phi\,[\text{rad}]$	$C\,[\text{T}]$
Training	$0.5$	$2\pi/1.4$	$-0.05+\pi/2$	$1.3$
	$0.45$	$2\pi/1.5$	$1.0+\pi/2$	$1.2$
	$0.42$	$2\pi/0.7$	$\pi/2+3$	$1.3$
	$0.48$	$2\pi/1.1$	$\pi/2+1.3$	$1.1$
	$0.6$	$2\pi/1.0$	$-1.0$	$1.25$
	$0.7$	$2\pi/1.1$	$-1.0$	$1.1$

Table 4: Magnetic field configurations and subdivision of snapshots into training, validation and test sets.

Figure 11: Temporal profiles of magnetic field of the three selected test cases.

Figure 12: Results for the temperature (first column), velocity (second column) and pressure (third column) for test case A. The first row displays the reference full-order solution while the second row shows the reconstruction produced by the SHRED model. The third row reports the absolute difference between the FOM and the SHRED.

Figure 13: Results for the temperature (first column), velocity (second column) and pressure (third column) for test case B. The first row displays the reference full-order solution while the second row shows the reconstruction produced by the SHRED model. The third row reports the absolute difference between the FOM and the SHRED.

Figure 14: Results for the temperature (first column), velocity (second column) and pressure (third column) for test case C. The first row displays the reference full-order solution while the second row shows the reconstruction produced by the SHRED model. The third row reports the absolute difference between the FOM and the SHRED.

Figure 15: Temporal behavior of relative $L^{2}$ -error of the SHRED reconstruction over time for temperature, velocity and pressure for the three test cases

Figure 16: Real temporal profile and SHRED estimation of the rescaled magnetic field in the three test cases.

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Surrogate models for nuclear fusion with parametric Shallow Recurrent Decoder Networks: applications to magnetohydrodynamics

M. Lo Verso, C. Introini, E. Cervi, L. Savoldi+2

Department of Energy, Politecnico di Milano, Milano, 20133, Italy · MATHEP Group, Dept. of Energy "Galileo Ferraris", Politecnico di Torino, Torino, Italy · Autodesk Research, 6 Agar Street, London UK+1

Magnetohydrodynamics Neural Surrogate Modeling Reduced-Order Modeling Computational Fluid Dynamics Recurrent Neural Networks

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Magnetohydrodynamic (MHD) effects play a key role in the design and operation of nuclear fusion systems, where electrically conducting fluids (such as liquid metals or molten salts in reactor blankets) interact with magnetic fields of varying intensity and orientation, which affect the resulting flow. The numerical resolution of MHD models involves highly nonlinear multiphysics systems of equations and can become computationally expensive, particularly in multi-query, parametric, or real-time contexts. This work investigates a fully data-driven framework for MHD state reconstruction that combines dimensionality reduction via Singular Value Decomposition (SVD) with the SHallow REcurrent Decoder (SHRED), a neural network architecture designed to recover the full spatio-temporal state from sparse time-series measurements of a limited number of observables. The methodology is applied to a parametric MHD test case involving compressible lead-lithium flow in a stepped channel subjected to thermal gradients and magnetic fields spanning a broad range of intensities. To improve efficiency, the full-order dataset is first compressed using SVD, yielding a reduced representation used as reference truth for training. Only temperature measurements from three sensors are provided as input, while the network reconstructs the full fields of velocity, pressure, and temperature. To assess robustness with respect to sensor placement, thirty randomly generated sensor configurations are tested in ensemble mode. Results show that SHRED accurately reconstructs the full MHD state even for magnetic field intensities not included in the training set. These findings demonstrate the potential of SHRED as a computationally efficient surrogate modeling strategy for fusion-relevant multiphysics problems, enabling low-cost state estimation with possible applications in real-time monitoring and control.

Abstract

Magnetohydrodynamics

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