cs.LGApr 3, 2026

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

Read PDF
Ask AI
Cite

Copy citation

DOINot available for this paper

Abstract

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 7.85×10−57.85 \times 10^{-5} m and a velocity mean absolute error of 3.21×10−33.21 \times 10^{-3} m/s under the nominal condition with Δt=1.0Δt = 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,…v_{1},v_{2},\ldots for FP velocities; h6h_{6} for CV water heights) is predicted by a dedicated subnetwork that takes time tt 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\Delta h , vov_{o} , and tt as inputs and predicts the velocity vv , from which ∂v/∂t\partial v/\partial t is computed via automatic differentiation. The momentum residual Lmomentum\mathcal{L}_{\mathrm{momentum}} 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)(\Delta h,\,v_{o})MAE (m/s)MSE (m/s) 2
(1.0,  0)(1.0,\;0)7.30×10−37.30\times 10^{-3}4.19×10−44.19\times 10^{-4}
(2.0,  3.0)(2.0,\;3.0)6.71×10−36.71\times 10^{-3}5.88×10−45.88\times 10^{-4}
(1.0,  6.0)(1.0,\;6.0)4.10×10−34.10\times 10^{-3}1.70×10−41.70\times 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 vnv_{n} , and the velocity vn+1v_{n+1} in each FP is obtained through the parametrized NA-PINN inference given Δh\Delta h , vov_{o} , and Δt\Delta t .
Figure 6: Verification under the nominal initial condition ( Hinit=[2,  0,  0,  0,  0,  0]H_{\mathrm{init}}=[2,\;0,\;0,\;0,\;0,\;0] m). Each row corresponds to a time step ( Δt=0.2\Delta t=0.2 , 0.50.5 , 1.01.0 s from top to bottom). Left column: CV water levels; right column: FP velocities. Solid: hybrid PINN–FDM; dashed: reference FDM.
 Water level hh (m)Velocity vv (m/s)
Δt\Delta t (s)MAEMSEMAEMSE
0.29.32×10−59.32\times 10^{-5}1.78×10−81.78\times 10^{-8}3.08×10−33.08\times 10^{-3}3.20×10−53.20\times 10^{-5}
0.51.01×10−41.01\times 10^{-4}2.26×10−82.26\times 10^{-8}5.55×10−35.55\times 10^{-3}1.02×10−41.02\times 10^{-4}
1.07.85×10−57.85\times 10^{-5}9.92×10−99.92\times 10^{-9}3.21×10−33.21\times 10^{-3}3.59×10−53.59\times 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\Delta t=0.2 sΔt=0.5\Delta t=0.5 sΔt=1.0\Delta t=1.0 s
Reference FDM solver
time (s)0.1790.0750.036
Hybrid PINN–FDM framework
Total inference (s)5.0512.0431.007
Speedup ratio0.04x0.04x0.04x
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\Delta t=1.0 s). Top row: Case 1 ( Hinit=[1.5,  0.5,  0,  0,  0,  0]H_{\mathrm{init}}=[1.5,\;0.5,\;0,\;0,\;0,\;0] m); bottom row: Case 2 ( Hinit=[1.0,  0.5,  0.5,  0,  0,  0]H_{\mathrm{init}}=[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 hh (m)Velocity vv (m/s)
CaseHinitH_{\mathrm{init}} (m)MAEMSEMAEMSE
1[1.5,  0.5,  0,  0,  0,  0][1.5,\;0.5,\;0,\;0,\;0,\;0]9.43×10−59.43\times 10^{-5}1.77×10−81.77\times 10^{-8}3.24×10−33.24\times 10^{-3}3.34×10−53.34\times 10^{-5}
2[1.0,  0.5,  0.5,  0,  0,  0][1.0,\;0.5,\;0.5,\;0,\;0,\;0]9.35×10−59.35\times 10^{-5}1.64×10−81.64\times 10^{-8}3.57×10−33.57\times 10^{-3}3.70×10−53.70\times 10^{-5}
3[1.3,  0.7,  0,  0,  0,  0][1.3,\;0.7,\;0,\;0,\;0,\;0]9.32×10−59.32\times 10^{-5}1.71×10−81.71\times 10^{-8}3.37×10−33.37\times 10^{-3}3.51×10−53.51\times 10^{-5}
4[0.5,  0.5,  0.5,  0.5,  0,  0][0.5,\;0.5,\;0.5,\;0.5,\;0,\;0]9.29×10−59.29\times 10^{-5}1.49×10−81.49\times 10^{-8}4.17×10−34.17\times 10^{-3}4.54×10−54.54\times 10^{-5}
5[1.0,  0.5,  0.3,  0.2,  0,  0][1.0,\;0.5,\;0.3,\;0.2,\;0,\;0]9.05×10−59.05\times 10^{-5}1.50×10−81.50\times 10^{-8}3.50×10−33.50\times 10^{-3}3.64×10−53.64\times 10^{-5}
Table 4: Prediction errors across five initial condition cases ( Δt=1.0\Delta t=1.0 s). Errors are computed over all CVs (water level) and all FLs (velocity).
More in this topic

Physics-Based Simulation

Physics-based simulation models physical systems with equations and numerical solvers, increasingly augmented by learned models for speed and control.

See all 228 papers

Explore similar work

CardsList
Jun 6, 2026

Overcoming the Limits of Finite Difference Method; Physics-Informed Neural Network for Noisy High-Dimensional Heat Diffusion

High-dimensional transient heat diffusion under noisy boundary conditions exposes a fundamental limitation of classical numerical methods: accuracy degrades catastrophically where physical noise is unavoidable. This paper presents a Physics-Informed Neural Network (PINN) framework as a systematic solution to this problem across one, two, and three spatial dimensions, establishing clear operational regimes that redefine solver selection in noisy thermal systems. Under 20% boundary noise in 3D, PINN sustains approximately 91% accuracy while Finite Difference Method (FDM) collapses to 36%, a clear decisive advantage. This is further confirmed in a physical copper thermal system, where PINN reduces boundary reconstruction error by 3.3 times under realistic noise conditions. This noise resilience is accompanied by a dimensionality-driven efficiency crossover: PINN requires fewer spacetime nodes than FDM in 3D while achieving superior accuracy, exposing the true cost of classical discretization at scale. These findings reframe solver selection: the decisive axis is not accuracy alone, but noise exposure and dimensionality jointly. When noise and dimensionality are both high, the classical solver paradigm is insufficient; this work provides the foundation to justify PINN as the operational standard in such regimes.

Shreesh Bhattarai, Harish Chandra Bhandari

Department of Mathematics, Kathmandu University Dhulikhel, Nepal

PDE SolvingPhysics-Informed ML
AskPDF
Jul 29, 2026

Comparison of a Parametric Physics-Informed Neural Network and a Tensorial Reduced-Order Model for the Shallow-Water Dam-Break Problem

We develop two parametric data-driven reduced models: a physics-informed neural network (PINN) and a non-intrusive tensorial reduced-order model (TROM), and apply both approaches to the parametrized one-dimensional shallow-water dam-break problem. Neither reduced model requires time integration: both learn a direct parameter-to-solution map from space, time, and dam-break parameters to the physical state, with the PINN providing predictions at arbitrary times and the TROM reconstructing solutions at the stored snapshot times. In addition, we demonstrate that it is essential to introduce shock-aware collocation to improve the robustness of the PINN model.

Anton Myshak, Md Rezwan Bin Mizan, Ilya Timofeyev

Dept. of Mathematics, University of Houston, Houston, TX 77204

PDE Surrogate ModelingReduced-Order ModelingPhysics-Informed MLComputational Fluid Dynamics
AskPDF
May 9, 2026

Finite Volume-Informed Neural Network Framework for 2D Shallow Water Equations: Rugged Loss Landscapes and the Importance of Data Guidance

Physics-informed neural networks (PINNs) are a simple surrogate-modelling paradigm for partial differential equations, but their standard strong-form residual formulation is ill suited to the shallow water equations (SWE). It cannot enforce local conservation, handle discontinuities, or leverage the boundary-conforming unstructured meshes used in real-world applications. We introduce ``Data-Guided FVM-PINN'', a framework that replaces the strong-form residual with a differentiable, well-balanced Roe Riemann-solver finite-volume (FVM) loss evaluated on unstructured meshes. The major finding is that physics-only FVM-PINN training often fails on realistic 2D problems: the network collapses to a trivial low-momentum state that nearly satisfies the FVM-PINN residual but bears no resemblance to the true flow. A loss-landscape diagnostic shows that the FVM-PINN loss at zero momentum is only about 7×7\times larger than at the trained solution, a shallow basin that an ordinary optimizer falls into; adding even sparse data turns this into a 310×310\times separation, breaking the degeneracy. On a 2D block-in-channel benchmark, just 200200 random velocity measurements drop the velocity-field L2L_2 error by 22×22\times versus physics-only; 5050 measurements still deliver a 7×7\times reduction. A controlled ablation isolates the contribution of the FVM-PINN loss: it reduces velocity-field L2L_2 by \sim$$23\% in the sparse-data regime and is essentially neutral when dense reference data is available. On a real-world Savannah River reach (13061306 cells, 36003600~s simulation, five Manning zones), the framework constructs an accurate surrogate from SRH-2D anchor data, with time-window decomposition reducing error monotonically via progressive initial-condition handoff.

Xiaofeng Liu

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

PDE SolvingPDE Surrogate ModelingPhysics-Informed MLComputational Fluid Dynamics
AskPDF