Coupling a parameterized PINN with FDM by node assignment yields water-level MAE of about 7.85×10⁻⁵ m and velocity MAE of about 3.21×10⁻³ m/s in a six-tank draining case without retraining

Proposes the first data-free surrogate model for a nuclear thermal-hydraulic system code: the parameterized NA-PINN takes the water-level difference Δh, initial velocity v₀, and time t as inputs, learns a solution manifold over the parameter space, and a single trained network predicts velocity across all flow path (FP) nodes without retraining or simulation data. The prior NA-PINN, like standard PINNs, was valid only for a single fixed scenario and required complete retraining for any parameter change; data-driven surrogates offer parametric flexibility but depend on large volumes of simulation data. This work extends NA-PINN to a parameterized setting while retaining data-free training. Compared against reference FDM solutions for three representative input conditions (Δh, v₀) = (1.0, 0), (2.0, 3.0), and (1.0, 6.0), with MAE of O(10⁻³) m/s and MSE of O(10⁻⁴) (m/s)², covering quiescent start-up, simultaneous driving head and initial momentum, and over-velocity deceleration.

Proposes a node-assigned hybrid coupling strategy (the P2F method): the parameterized PINN handles the nonlinear momentum equation via a single forward pass, while the FDM solver advances the mass conservation equation at each time step, enforcing exact discrete mass conservation at every step. Existing hybrid PINN-numerical frameworks either replace the differential operator computation within PINNs or perform spatial domain decomposition, but no study had realized a coupling in which a data-free PINN surrogate and a conventional numerical solver are assigned to distinct physical nodes and advance together within a time-marching loop. In the six-tank gravity-driven draining scenario under the nominal initial condition (only CV01 filled to 2.0 m), the framework matches the reference FDM solution at Δt = 0.2, 0.5, and 1.0 s, with water level MAE of O(10⁻⁵) m and velocity MAE within 3–6×10⁻³ m/s.

The framework is robust to time-step variation and generalizes across initial conditions: accuracy is consistent for time steps from 0.2 to 1.0 s, and it generalizes to five distinct initial water-level distributions without retraining. Errors do not increase monotonically with Δt; the Δt = 1.0 s case yields errors comparable to Δt = 0.2 s. Across the five initial conditions (including Case 4, where equal levels in four tanks give zero initial driving head across FL01–FL03), water level MAE stays at 9.05–9.43×10⁻⁵ m and velocity MAE ranges from 3.24×10⁻³ to 4.17×10⁻³ m/s. All five initial conditions are compared against reference FDM solutions at Δt = 1.0 s, with overall uniformity of error magnitudes; Case 4 shows the highest velocity error but the increase is modest and water level accuracy is unaffected.

A hard constraint embeds the initial condition directly into the network output (v̂(t) = v₀ + t·NNθ), reducing the training loss to a single physics-based residual term and eliminating the multi-objective balancing problem. The conventional soft-constraint approach appends penalty terms and requires careful tuning of weighting coefficients, introducing competing objectives; the hard constraint satisfies v̂(0) = v₀ exactly for any network parameters, letting the optimizer focus exclusively on minimizing the physics-based residual. Training uses a fixed collocation set with boundary-enriched sampling (a controlled fraction of points at h = 0 and v₀ = 0), together with a piecewise learning rate schedule, gradient clipping, and early stopping based on validation loss.

The framework targets surrogate modeling for nuclear thermal-hydraulic system codes such as MELCOR's CVH/FP module, in idealized settings with open tanks, fixed flow direction, and sequential upwind-scheme coupling. It enables a single trained parameterized NA-PINN to be reused across all flow paths within the trained parameter range without simulation data or retraining, and it is directly compatible with the node-based FDM time-marching structure of existing system codes. For nuclear engineering researchers working on severe accident analysis, parametric studies, and uncertainty quantification, the method offers a path to replace the iterative nonlinear momentum solve with a single forward pass while preserving discrete mass conservation.

Verification is limited to an idealized open-tank scenario in which the fixed flow direction permits sequential upwind coupling; extension to closed systems, non-negligible pressure differentials, bidirectional flow, and the corresponding matrix-based implicit coupling remains to be done. The implementation currently covers only the CVH/FP module, and multi-physics scalability after incorporating the Heat Structure (HS) and Radionuclide (RN) modules remains to be verified. Under the simplified open-tank equations the hybrid framework is approximately 25× slower than the reference FDM solver, and systematic benchmarking under progressively more complex equations is needed to identify when it becomes cost-competitive. In addition, interpolation beyond the training parameter range remains limited, so the training range must cover the expected operating conditions.