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
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 ] .
ρ0
9806kg/m3
κ
20.93W m−1K−1
pext
105Pa
μ
1.93×10−3Pa⋅s
c
189.5J kg−1K−1
Nh
112640
μB
1.26×10−6H/m
u0
0.01m/s
L
0.07m
σ
7.82×105Ω−1m−1
T0
600K
a
0.02m
β
1.3×10−4K−1
Tpipe
560K
r
0.005m
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 Bx=0.75T at time t=2s . 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 Bx=1.85T at time t=2s . 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 Bx=2.5T at time t=2s . 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 L2 -error of the SHRED reconstruction over time for temperature, velocity and pressure for the cases with Bx=0.75T (a), Bx=1.85T (b) and Bx=2.5T (c).
Snapshots
Bx
By
∣Btot∣
α
Training
1.0T
0.2T
1.02T
11.31∘
1.4T
0.35T
1.44T
14.04∘
1.8T
0.55T
1.88T
16.99∘
2.0T
0.7T
2.12T
19.29∘
Validation
1.2T
0.3T
1.24T
14.04∘
Test
1.6T
0.45T
1.66T
15.71∘
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 Bx=1.60T and By=0.45T at time t=2s . 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 L2 -error of the SHRED reconstruction over time for temperature, velocity and pressure for the test case with Bx=1.6T and By=0.45T .
Set
A[T]
ω[1/s]
ϕ[rad]
C[T]
Training
0.5
2π/1.4
−0.05+π/2
1.3
0.45
2π/1.5
1.0+π/2
1.2
0.42
2π/0.7
π/2+3
1.3
0.48
2π/1.1
π/2+1.3
1.1
0.6
2π/1.0
−1.0
1.25
0.7
2π/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 L2 -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.
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.
M. Lo Verso, C. Introini, E. Cervi +3
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
State estimation in magnetohydrodynamic flows is critical for real-time monitoring of liquid metal blankets in tokamak fusion reactors. Due to the multiphysics nature of these phenomena, high-fidelity simulations are computationally prohibitive for real-time applications. This work investigates a data- driven Reduced Order Model framework: the Shallow Recurrent Decoder (SHRED) coupled with Principal Component Analysis, to map sparse temperature measurements to the full thermo-hydraulic system's state. The major contribution of this work lies in the two-parameter analysis of a fully three-dimensional domain representative of the DEMO breeding blanket configuration. Here, the flow is subjected to an external magnetic field varying in direction and intensity and is hindered by two cylinders acting as a water-cooling system, which impose a temperature boundary condition on their surfaces. This double-parametric magnetic variation induces nonlinear transitions in the flow dynamics, ranging from chaotic behavior at low magnetic field intensities to laminarized regimes at high intensities, characterized by the formation of asymmetric side layers at an inclination angle of 30 degrees. SHRED reconstruction maintains a mean relative error of approximately 5% for the temperature, pressure, and velocity fields. This accuracy is maintained across both weak and strong magnetic fields, ranging from 0.075 T to 0.300 T, and for inclination angles from 5 to 30 degrees, reflecting its dominant toroidal component. These errors are only slightly larger than the lower error bound dictated by low-rank truncation. The results establish SHRED as a reliable state estimator for complex and realistic engineering applications involving completely unseen parametric scenarios and validate it as an accurate real-time state estimation technique suitable for online monitoring and control of real facilities.
Energy Department, CeSNEF - Nuclear Engineering Division, Politecnico di Milano, Milano · Autodesk Research, 6 Agar Street, London, WC2R 0QE, UK · Argonne National Laboratory, Lemont, IL, 60439, United States of America +1
Reconstructing spatially resolved plasma dynamics from few sensors is essential for diagnostics, reduced-order modelling and control, yet remains difficult because the sparse measurements incompletely constrain multiscale, regime-dependent degrees of freedom. The Shallow Recurrent Decoder (SHRED) partially addresses spatial sparsity by using measurement histories; however, its fully connected decoder provides no explicit mechanism for resolving spatial structure across scales or explicit parametric dependency. We introduce the Recurrent Multiscale Affine-modulated Inference Network (ReMAIN), which preserves SHRED's recurrent temporal encoding but replaces its decoder with a U-Net whose feature hierarchy is conditioned by the recurrent state through feature-wise linear modulation. The temporal representation supplies both a dense prior and scale-specific modulation throughout the U-Net. A parametric extension jointly embeds the operating condition and sensor history, enabling reconstruction to adapt as the governing dynamics change with operating regime. ReMAIN is first benchmarked against SHRED on six one-dimensional nonlinear PDEs representing diverse dynamics. Across all benchmarks, it reduces reconstruction errors on unseen trajectories and more faithfully resolves sharp transitions, localized extrema and fine-scale variations. The parameter-conditioned model is then demonstrated on a collisionless E×B plasma subject to perpendicular axial electric and radial magnetic fields, with the electric-field strength serving as the operating parameter. ReMAIN reconstructs the high-dimensional, multiscale plasma state and recovers its regime-dependent spatiotemporal dynamics at electric-field strengths withheld from training. Together, ReMAIN improves sparse-sensor full-state reconstruction and, through parameter conditioning, generalizes across plasma operating regimes.
Maryam Reza, Farbod Faraji
Independent Researcher · Department of Computing, Huxley Building, Imperial College London, London SW7 2RH, United Kingdom