A Numerical Method for Coupling Parameterized Physics-Informed Neural Networks and FDM for Advanced Thermal-Hydraulic System Simulation
Authors: Jeesuk Shin, Donggyun Seo, Sihyeong Yu, Joongoo Jeon
Organizations: 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
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.
Figures & tables
Figure 1: Architecture of the original NA-PINN [ 24 ] . Each nodal variable ( v1,v2,… for FP velocities; h6 for CV water heights) is predicted by a dedicated subnetwork that takes time t as input. Temporal derivatives are computed via automatic differentiation, and the weighted momentum and continuity residuals are summed to form the total loss.
Figure 2: Nodalization for the scenario model.
Figure 3: Architecture of the reformulated NA-PINN for a single FP node. The FCN takes Δh , vo , and t as inputs and predicts the velocity v , from which ∂v/∂t is computed via automatic differentiation. The momentum residual Lmomentum is evaluated and accumulated into the total loss.
Figure 4: Standalone PINN predictions versus reference FDM solutions for three input conditions. Solid: parameterized PINN; dashed: reference FDM.
(Δh,vo)
MAE (m/s)
MSE (m/s) 2
(1.0,0)
7.30×10−3
4.19×10−4
(2.0,3.0)
6.71×10−3
5.88×10−4
(1.0,6.0)
4.10×10−3
1.70×10−4
Table 1: Prediction errors of the standalone parameterized PINN for three input conditions.
Figure 5: Architecture of the P2F framework. At each time step, the water level in each CV is sequentially updated via the FDM-based mass conservation equation using the predicted velocity vn , and the velocity vn+1 in each FP is obtained through the parametrized NA-PINN inference given Δh , vo , and Δt .
Figure 6: Verification under the nominal initial condition ( Hinit=[2,0,0,0,0,0] m). Each row corresponds to a time step ( Δt=0.2 , 0.5 , 1.0 s from top to bottom). Left column: CV water levels; right column: FP velocities. Solid: hybrid PINN–FDM; dashed: reference FDM.
Water level h (m)
Velocity v (m/s)
Δt (s)
MAE
MSE
MAE
MSE
0.2
9.32×10−5
1.78×10−8
3.08×10−3
3.20×10−5
0.5
1.01×10−4
2.26×10−8
5.55×10−3
1.02×10−4
1.0
7.85×10−5
9.92×10−9
3.21×10−3
3.59×10−5
Table 2: Prediction errors under the nominal initial condition for three time steps. Errors are computed over all CVs (water level) and all FLs (velocity).
Δt=0.2 s
Δt=0.5 s
Δt=1.0 s
Reference FDM solver
time (s)
0.179
0.075
0.036
Hybrid PINN–FDM framework
Total inference (s)
5.051
2.043
1.007
Speedup ratio
0.04x
0.04x
0.04x
Table 3: Computational cost comparison between the reference FDM solver and the hybrid PINN–FDM framework under the nominal initial condition.
Figure 7: Generalization to unseen initial conditions ( Δt=1.0 s). Top row: Case 1 ( Hinit=[1.5,0.5,0,0,0,0] m); bottom row: Case 2 ( Hinit=[1.0,0.5,0.5,0,0,0] m). Left: CV water levels; right: FP velocities. Solid: hybrid PINN–FDM; dashed: reference FDM.
Water level h (m)
Velocity v (m/s)
Case
Hinit (m)
MAE
MSE
MAE
MSE
1
[1.5,0.5,0,0,0,0]
9.43×10−5
1.77×10−8
3.24×10−3
3.34×10−5
2
[1.0,0.5,0.5,0,0,0]
9.35×10−5
1.64×10−8
3.57×10−3
3.70×10−5
3
[1.3,0.7,0,0,0,0]
9.32×10−5
1.71×10−8
3.37×10−3
3.51×10−5
4
[0.5,0.5,0.5,0.5,0,0]
9.29×10−5
1.49×10−8
4.17×10−3
4.54×10−5
5
[1.0,0.5,0.3,0.2,0,0]
9.05×10−5
1.50×10−8
3.50×10−3
3.64×10−5
Table 4: Prediction errors across five initial condition cases ( Δt=1.0 s). Errors are computed over all CVs (water level) and all FLs (velocity).
Department of Civil and Environmental Engineering, Pennsylvania State University, University Park, PA, USA · Institute for Computational and Data Sciences, Pennsylvania State University, University Park, PA, USA