Residual Attention Physics-Informed Neural Networks for Robust Multiphysics Simulation of Steady-State Electrothermal Energy Systems
Authors: Yuqing Zhou, Ze Tao, Fujun Liu
Organizations: Nanophotonics and Biophotonics Key Laboratory of Jilin Province, School of Physics, Changchun University of Science and Technology, Changchun, 130022, P.R. China
Efficient thermal management and precise field prediction are critical for the design of advanced energy systems, including electrohydrodynamic transport, microfluidic energy harvesters, and electrically driven thermal regulators. However, the steady-state simulation of these electrothermal coupled multiphysics systems remains challenging for physics-informed neural computation due to strong nonlinear field coupling, temperature-dependent coefficient variability, and complex interface dynamics. This study proposes a Residual Attention Physics-Informed Neural Network (RA-PINN) framework for the unified solution of coupled velocity, pressure, electric-potential, and temperature fields. By integrating a unified five-field operator formulation with residual-connected feature propagation and attention-guided channel modulation, the proposed architecture effectively captures localized coupling structures and steep gradients. We evaluate RA-PINN across four representative energy-relevant benchmarks: constant-coefficient coupling, indirect pressure-gauge constraints, temperature-dependent transport, and oblique-interface consistency. Comparative analysis against Pure-MLP, LSTM-PINN, and pLSTM-PINN demonstrates that RA-PINN achieves superior accuracy, yielding the lowest MSE, RMSE, and relative L2 errors across all scenarios. Notably, RA-PINN maintains high structural fidelity in interface-dominated and variable-coefficient settings where conventional PINN backbones often fail. These results establish RA-PINN as a robust and accurate computational framework for the high-fidelity modeling and optimization of complex electrothermal multiphysics in sustainable energy applications.
Figures & tables
Figure 1: Workflow of the RA-PINN solver used in this study. The schematic organizes the method into six coupled parts: governing equations and computational domain, adaptive residual sampling, residual-attention PINN mapping, residual adaptation, gradient-based optimization, and prediction/validation. The uploaded schematic uses generic space-time notation; in the present steady-state problem the network input reduces to the spatial coordinate (x,y) and the outputs reduce to the five coupled fields u^ , v^ , p^ , ϕ^ , and T^ .
Case
Model
Field
MSE
RMSE
MAE
Relative L2
Case 1
RA-PINN
u
8.283×10−7
9.101×10−4
7.095×10−4
2.661×10−3
v
7.918×10−7
8.898×10−4
6.665×10−4
3.256×10−3
p
7.026×10−7
8.382×10−4
6.588×10−4
4.756×10−3
T
1.374×10−6
1.172×10−3
8.733×10−4
2.221×10−3
phi
8.446×10−7
9.190×10−4
5.973×10−4
3.279×10−3
Avg.
9.083×10−7
9.459×10−4
7.011×10−4
3.235×10−3
Table 1: Fieldwise and average error table for all four benchmark cases. Each model reports the five physical fields u , v , p , T , and ϕ , together with an average row over the five fields. The minimum value for each case, field, and metric appears in bold.
Case
RA-PINN
LSTM-PINN
pLSTM-PINN
Pure-MLP
Fastest
Case 1
24.01
14.29
1.09
1.40
pLSTM-PINN
Case 2
24.67
14.76
4.19
3.17
Pure-MLP
Case 3
39.81
11.83
4.28
4.45
pLSTM-PINN
Case 4
38.35
18.35
9.30
14.76
pLSTM-PINN
Table 2: Training time in hours converted from the logged time-cost files.
Figure 2: Uploaded benchmark comparison for Case 1. The top strip reports the loss-history comparison, the five rows correspond to p , u , v , ϕ , and T , the left block reports the benchmark and predicted fields, and the right block reports the absolute-error maps of the four learned solvers.
Figure 3: Uploaded benchmark comparison for Case 2. This benchmark replaces direct pressure anchoring with a zero-mean gauge constraint, so the figure tests how each network reconstructs the five coupled fields under indirect pressure identifiability.
Figure 4: Uploaded benchmark comparison for Case 3. This benchmark activates temperature-dependent transport coefficients, so the figure directly tests how well each network resolves variable-coefficient coupling across the five physical fields.
Figure 5: Uploaded benchmark comparison for Case 4. The benchmark contains an oblique material interface, so the figure highlights interface-sensitive reconstruction quality by placing the learned absolute-error maps next to the predicted multiphysics fields.
Physics-Informed Neural Networks present a novel approach in SciML that integrates physical laws in the form of partial differential equations directly into the NN through soft constraints in the loss function. This work studies the application of PINNs to solve a one dimensional coupled electro-elastodynamic system modeling linear piezoelectricity in stress-charge form, governed by elastodynamic and electrodynamic equations. Our simulation employs a feedforward architecture, mapping space-time coordinates to mechanical displacement and electric potential. Our PINN model achieved global relative L2 errors of 2.34 and 4.87 percent for displacement and electric potential respectively. The results validate PINNs as effective mesh free solvers for coupled time-dependent PDE systems, though challenges remain regarding error accumulation and stiffness in coupled eigenvalue systems.
Suhas Suresh Bharadwaj, Reuben Thomas Thovelil
Department of Electrical and Electronics Engineering, BITS Pilani, Dubai Campus, Dubai, United Arab Emirates
Multi-domain physics-informed neural networks (PINNs) flexibly model medium-specific representations to solve fluid--solid conjugate heat transfer (CHT). However, standard multi-domain PINNs enforce governing equations and interface conditions on separately sampled domain supports, which can yield plausible temperature fields but inaccurate end-to-end energy transfer and outlet temperatures. We propose Cova-PINN, a multi-domain PINN framework that aligns conservation support with thermal interaction paths in complex geometries. Cova-PINN jointly optimizes cross-domain composite control-volume balances at the local scale and paired-wall closure at the global exchanger scale. We evaluate Cova-PINN on four triply periodic minimal surface (TPMS) heat exchangers and a geometrically distinct DualMS design against CHT-specific, optimization-oriented, and complex-geometry PINN baselines under a common protocol. Relative to the closest baseline, MUSA-PINN-CHT, Cova-PINN reduces average outlet-temperature and device-level closure errors across the four TPMS topologies by 37.7% and 60.2%, respectively, while also improving full-field and heat-duty accuracy, with consistent gains on DualMS.
Severe accident analysis using system-level codes such as MELCOR is indispensable for nuclear safety assessment, yet the computational cost of repeated simulations poses a significant bottleneck for parametric studies and uncertainty quantification. Existing surrogate models accelerate these analyses but depend on large volumes of simulation data, while physics-informed neural networks (PINNs) enable data-free training but must be retrained for every change in problem parameters. This study addresses both limitations by developing the Parameterized PINNs coupled with FDM (P2F) method, a node-assigned hybrid framework for MELCOR's Control Volume Hydrodynamics/Flow Path (CVH/FP) module. In the P2F method, a parameterized Node-Assigned PINN (NA-PINN) accepts the water-level difference, initial velocity, and time as inputs, learning a solution manifold so that a single trained network serves as a data-free surrogate for the momentum conservation equation across all flow paths without retraining. This PINN is coupled with a finite difference method (FDM) solver that advances the mass conservation equation at each time step, ensuring exact discrete mass conservation while replacing the iterative nonlinear momentum solve with a single forward pass. Verification on a six-tank gravity-driven draining scenario yields a water level mean absolute error of 7.85×10−5 m and a velocity mean absolute error of 3.21×10−3 m/s under the nominal condition with Δt=1.0 s. The framework maintains consistent accuracy across time steps ranging from 0.2 to 1.0 s and generalizes to five distinct initial conditions, all without retraining or simulation data. This work introduces a numerical coupling methodology for integrating parameterized PINNs with FDM within a nuclear thermal-hydraulic system code framework.
Jeesuk Shin, Donggyun Seo, Sihyeong Yu +1
Division of Advanced Nuclear Engineering, Pohang University of Science and Technology, Pohang-si, Republic of Korea · Mueunjae School of Undergraduate Studies, Pohang University of Science and Technology, Pohang-si, Republic of Korea · Department of Quantum System Engineering, Jeonbuk National University, Jeonju-si, Republic of Korea