A Numerical Method for Coupling Parameterized Physics-Informed Neural Networks and FDM for Advanced Thermal-Hydraulic System Simulation

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 $7.85 \times 10^{-5}$ m and a velocity mean absolute error of $3.21 \times 10^{-3}$ 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.

• The paper presents a hybrid method coupling parameterized PINN surrogates with finite difference solvers for efficient thermal-hydraulic system simulation.
• It enforces hard constraints and alternates node updates to mitigate long-horizon error accumulation while maintaining numerical stability.
• The approach achieves high-fidelity results across various initial conditions, reducing computational overhead for nuclear safety analyses.

Authoritative Summary of "A Numerical Method for Coupling Parameterized Physics-Informed Neural Networks and FDM for Advanced Thermal-Hydraulic System Simulation" (2604.02663)

Problem Motivation and Context

The paper addresses computational bottlenecks in severe accident analysis for nuclear power plant safety, particularly in simulations using system-level codes such as MELCOR. Traditional approaches require extensive repeated runs for parametric studies and uncertainty quantification, incurring substantial computational costs. Existing data-driven surrogate models improve efficiency but rely heavily on simulation data, while conventional physics-informed neural networks (PINNs) offer data-free training yet lack parametric generality and require retraining for every variation in initial or boundary conditions.

Two critical gaps are identified:

1. The lack of a data-free surrogate for nuclear thermal-hydraulic system codes with flexible parametric generalization.
2. The absence of node-wise hybrid numerical methods coupling data-free PINN surrogates and conventional solvers in a manner robust to long-horizon error accumulation.

Methodological Innovation

The proposed "Parameterized PINNs coupled with FDM (P2F)" method introduces a novel node-assigned hybrid framework for MELCOR's CVH/FP module. The central innovation is a parameterized Node-Assigned PINN (NA-PINN), which serves as a data-free surrogate for the momentum conservation equation across all flow paths. This PINN accepts water-level difference, initial velocity, and time as inputs, learning a joint solution manifold over the relevant parameter space. Only a single training phase is required, after which the PINN generalizes across different scenarios without retraining or dependence on simulation data.

The hybrid coupling operates as follows:

• The parameterized PINN solves the nonlinear momentum equation for each flow path in a single inference pass.
• A conventional finite difference method (FDM) solver advances the mass conservation equation, enforcing exact discrete mass conservation at every time step.
• The procedure is embedded in a time-marching loop, with the PINN and FDM solvers alternately updating different physical nodes, thereby preventing long-horizon error growth and maintaining numerical stability.

The architecture integrates hard constraints on initial conditions directly into the network output, removing the need for multi-objective loss balancing and improving training stability.

Numerical Results

Verification is performed on a six-tank gravity-driven draining scenario:

• The water level mean absolute error (MAE) is $7.85 \times 10^{-5}$ m, and velocity MAE is $3.21 \times 10^{-3}$ m/s with $\Delta t = 1.0$ s.
• The framework maintains accuracy across time steps ranging from $0.2$ to $1.0$ s.
• The trained PINN generalizes to five distinct initial condition configurations without retraining—water level MAE consistently remains at $O(10^{-5})$ m and velocity MAE at $O(10^{-3})$ m/s.

Notably, the parameterized NA-PINN achieves standalone surrogate performance indistinguishable from iterative FDM solutions, with stable prediction in diverse flow regimes and transient behaviors.

The current implementation, in the tested scenario, is approximately 25× slower than the reference FDM solver due to neural inference overhead, but this cost is fixed per time step irrespective of nonlinear complexity; thus, potential advantages may emerge with more challenging equations or large-scale implicit couplings.

Implications and Theoretical Impact

This work establishes the first data-free surrogate framework for thermal-hydraulic system codes, requiring no simulation data for training and no retraining for new parameter settings. The hybrid P2F approach is directly compatible with the FDM-based structure of prevalent system codes such as MELCOR, facilitating seamless integration and enabling practical acceleration in parametric studies or uncertainty quantification.

Practically, this framework could substantially reduce the computational cost in safety assessment pipelines, especially as complexity increases or as physical modules become more tightly coupled. The parameterized PINN schema enables generalization across a wide range of thermohydraulic scenarios, supporting scalable surrogate modeling and rapid exploration of operational uncertainties.

Theoretically, the node-assigned hybrid coupling avoids long-horizon PINN error accumulation and provides a robust strategy for combining data-free neural surrogates with discrete solvers. The methodology offers a blueprint for future work in multi-physics simulation, modular hybridization, and parameterized neural surrogate design, both within and beyond nuclear engineering.

Future Directions and Speculation

Open directions include:

• Extension to closed-system severe accident scenarios, non-negligible pressure couplings, and full matrix-based implicit velocity solutions consistent with MELCOR’s numerical scheme.
• Integration with additional MELCOR modules (e.g., Heat Structure and Radionuclide) for multi-physics simulation.
• Benchmarking of computational performance for progressively complex models, identifying cross-over points for hybrid PINN-FDM efficiency.
• Systematic exploration of the representational limits and extrapolation behaviors of parameterized PINNs in wider operational parameter spaces.

The framework’s scalable data-free surrogate capacity positions it as a candidate for broader AI acceleration in multi-domain physics-based codes, facilitating real-time simulation and rapid risk analysis in next-generation nuclear safety systems.

Conclusion

The paper introduces and validates a node-assigned hybrid numerical strategy coupling parameterized PINN surrogates and FDM solvers for advanced thermal-hydraulic system simulation. The methodology achieves data-free training, parametric flexibility, robust long-horizon stability, and direct compatibility with existing system code structures. Numerical results substantiate high-fidelity surrogate performance and robust generalization. Theoretical and practical implications extend to scalable simulation workflows and future AI-physics hybrid design paradigms.