Parameterized PINNs with FDM (P2F)

• Parameterized PINNs Coupled with FDM (P2F) is a hybrid framework that uses a neural surrogate for momentum and an FDM solver for mass conservation.
• The method accelerates thermal-hydraulic simulations by replacing iterative nonlinear solves with a single PINN forward pass while preserving conservation laws.
• P2F generalizes across varying operating conditions, offering robust, accurate predictions without retraining for new initial conditions.

Searching arXiv for the cited papers to ground the article in current records. arXiv search: (Shin et al., 3 Apr 2026) Parameterized PINNs coupled with FDM MELCOR CVH/FP Parameterized PINNs coupled with FDM (P2F) is a hybrid numerical framework in which a parameterized physics-informed neural network (PINN) is coupled to a finite difference method (FDM) time-marching solver so that a single trained network can act as a data-free surrogate over a family of operating conditions while the discrete solver retains conservation properties. In the formulation introduced for MELCOR’s Control Volume Hydrodynamics/Flow Path (CVH/FP) module, the parameterized Node-Assigned PINN (NA-PINN) learns the momentum response manifold as a function of water-level difference, initial velocity, and time, and an FDM solver advances the mass conservation equation at each time step, thereby preserving exact discrete mass conservation while replacing the iterative nonlinear momentum solve with a single forward pass (Shin et al., 3 Apr 2026). Related research places this method at the intersection of parameterized PINNs, data-free surrogate modeling, and PINN–discretization hybrids, but P2F is distinct both from parameterized PINNs that remain fully neural and from FDM-based PINNs in which finite-difference stencils replace automatic differentiation inside the PINN loss (Borrel-Jensen et al., 2021, Lim et al., 25 Feb 2026, Jangir et al., 4 Feb 2026).

1. Definition and conceptual scope

P2F was developed to address two limitations identified in accelerated thermal-hydraulic simulation. First, data-driven surrogates can be fast and parametric, but they require large amounts of MELCOR-generated training data. Second, standard PINNs are data-free, but they are typically trained for one fixed scenario; if initial or boundary conditions change, they must be retrained (Shin et al., 3 Apr 2026).

A second issue is the long-horizon stability problem. Standalone PINNs can accumulate error over long time intervals, which is problematic because severe-accident simulations may span hours or days. P2F responds by combining a parameterized NA-PINN with a conventional FDM solver in a time-marching loop. The network is not used as a full-horizon predictor. Instead, it predicts one short step at a time, and the FDM component updates the state variables through the mass equation. This coupling is intended to provide no training data requirement, no retraining for new initial conditions within the trained range, exact discrete mass conservation, and stable short-step prediction inside a conventional solver loop (Shin et al., 3 Apr 2026).

The defining conceptual split is therefore not merely “PINN plus discretization,” but a role-specific decomposition of the governing physics. The PINN handles momentum, which is the nonlinear part and, in conventional MELCOR treatment, requires iterative solution. The FDM component handles mass conservation, which can be advanced once flow velocities are known. This division distinguishes P2F from a pure parameterized PINN, which would attempt to represent the full spatiotemporal solution directly, and from an FDM-PINN, in which the discrete stencil is embedded into the PINN loss rather than coupled as an external marching solver (Lim et al., 25 Feb 2026).

2. Parameterized NA-PINN formulation

The parameterized NA-PINN in P2F is assigned to a single flow-path node and accepts water-level difference $\Delta h$, initial velocity $v_0$, and time $t$ as inputs, returning the flow-path velocity $v(t)$. The surrogate map is written as

$$v = \mathrm{NN}_{\boldsymbol{\theta}}(\Delta h, v_0, t).$$

The inputs are normalized as

$$\bar{h} = \frac{\Delta h}{\Delta h_{\mathrm{train}}}, \qquad \bar{t} = \frac{t}{T}, \qquad \bar{v}_0 = \frac{v_0}{v_{0,\max}},$$

where $\Delta h_{\mathrm{train}}$ is the maximum training head difference, $T$ is the training time horizon, and $v_{0,\max}$ is the maximum initial velocity in training (Shin et al., 3 Apr 2026).

A notable design choice is the hard enforcement of the initial condition. Rather than adding an initial-condition penalty term to the loss, the method defines

$$\hat{v}(t) = v_0 + t \cdot \mathrm{NN}_{\boldsymbol{\theta}}(\bar{h}, \bar{t}, \bar{v}_0),$$

which guarantees

$v_0$0

exactly. Because the initial condition is imposed structurally, there is no IC loss term in training (Shin et al., 3 Apr 2026).

The governing residual used during training is the simplified momentum equation for each flow path,

$v_0$1

where $v_0$2 is the inertial length of the flow path, $v_0$3 is the gravity-driven head term, and $v_0$4 is the quadratic friction loss. The total loss is the mean squared momentum residual over collocation points,

$v_0$5

Training uses fixed collocation points sampled over $v_0$6, $v_0$7, and $v_0$8, with extra samples at $v_0$9 and $t$0 to better learn the zero-flow regime. The optimization procedure uses first-order gradient-based training, a piecewise learning-rate schedule, gradient clipping, and fixed validation with best-model retention (Shin et al., 3 Apr 2026).

3. Coupling mechanism with the finite difference solver

The P2F framework couples the parameterized NA-PINN to an FDM solver by assigning distinct numerical roles to the two components. The PINN handles the momentum equation, while the FDM solver advances the mass conservation equation. This is the central numerical architecture of the method (Shin et al., 3 Apr 2026).

At time level $t$1, the coupling procedure is:

1. compute the head difference $t$2 from the current control-volume water levels;
2. feed $t$3 into the trained NA-PINN;
3. obtain $t$4 in one forward pass;
4. use the predicted velocities in the FDM discretization of the mass equation to update water levels $t$5.

In the online hybrid simulation, the workflow is described in more detail as state evaluation, PINN inference, FDM mass update, and state advancement. During state evaluation, current head differences $t$6 are computed from control-volume water levels and local void fractions $t$7 are determined. During inference, each flow path receives the tuple $t$8 and returns an updated velocity. During the FDM update, those velocities advance each control volume using the discretized mass balance. The process is repeated until final time (Shin et al., 3 Apr 2026).

The principal numerical significance of this arrangement is that the mass equation is advanced with FDM so that the discrete mass balance is maintained at every step. The method therefore preserves exact discrete mass conservation, unlike a fully neural surrogate that might drift. At the same time, the iterative nonlinear momentum solve of a conventional FDM-based system code is replaced by one PINN inference per flow path per step (Shin et al., 3 Apr 2026).

A common misunderstanding is to treat P2F as a standard PINN with a modified residual evaluation. That characterization does not fit the method’s actual construction. In P2F, finite differences are used as a solver component for the mass update in a time-stepping loop, whereas the momentum equation is represented by a trained parameterized surrogate. By contrast, in FDM-PINN formulations the finite-difference residual itself becomes the PINN’s physics loss (Lim et al., 25 Feb 2026).

4. Governing equations and verification problem

The verification study for P2F uses a six-tank gravity-driven draining problem with six open tanks, denoted CV01–CV06, and five connecting flow paths, denoted FL01–FL05. Each tank has cross-sectional area $t$9 and height $v(t)$0; each flow path has diameter $v(t)$1 and length $v(t)$2; and the elevation drop between tanks is $v(t)$3. All tanks are open to atmosphere, water drains from the upper tank to the lower ones under gravity, and the case is idealized so that flow direction is fixed and sequential (Shin et al., 3 Apr 2026).

The mass conservation equation used in the CVH/FP setting is

$v(t)$4

and the momentum conservation equation is

$v(t)$5

For the simplified open-tank problem, the model is reduced so that the neural surrogate depends on the local $v(t)$6 and $v(t)$7 (Shin et al., 3 Apr 2026).

The standalone parameterized NA-PINN verification uses three representative inputs:

1. $v(t)$8,
2. $v(t)$9,
3. $$v = \mathrm{NN}_{\boldsymbol{\theta}}(\Delta h, v_0, t).$$0.

These are described as spanning quiescent start-up, moderate head with initial velocity, and an over-velocity deceleration regime. For the hybrid P2F simulation, the nominal initial condition is

$$v = \mathrm{NN}_{\boldsymbol{\theta}}(\Delta h, v_0, t).$$1

and the time-step study is performed at $$v = \mathrm{NN}_{\boldsymbol{\theta}}(\Delta h, v_0, t).$$2 s, $$v = \mathrm{NN}_{\boldsymbol{\theta}}(\Delta h, v_0, t).$$3 s, and $$v = \mathrm{NN}_{\boldsymbol{\theta}}(\Delta h, v_0, t).$$4 s (Shin et al., 3 Apr 2026).

5. Accuracy, generalization, and computational behavior

For the standalone parameterized NA-PINN surrogate of the momentum equation, the reported mean absolute errors are $$v = \mathrm{NN}_{\boldsymbol{\theta}}(\Delta h, v_0, t).$$5 for $$v = \mathrm{NN}_{\boldsymbol{\theta}}(\Delta h, v_0, t).$$6, $$v = \mathrm{NN}_{\boldsymbol{\theta}}(\Delta h, v_0, t).$$7 for $$v = \mathrm{NN}_{\boldsymbol{\theta}}(\Delta h, v_0, t).$$8, and $$v = \mathrm{NN}_{\boldsymbol{\theta}}(\Delta h, v_0, t).$$9 for $$\bar{h} = \frac{\Delta h}{\Delta h_{\mathrm{train}}}, \qquad \bar{t} = \frac{t}{T}, \qquad \bar{v}_0 = \frac{v_0}{v_{0,\max}},$$0. The standalone surrogate therefore achieves $$\bar{h} = \frac{\Delta h}{\Delta h_{\mathrm{train}}}, \qquad \bar{t} = \frac{t}{T}, \qquad \bar{v}_0 = \frac{v_0}{v_{0,\max}},$$1 accuracy on the stated verification points (Shin et al., 3 Apr 2026).

Under the nominal initial condition in the hybrid P2F setting, the water level MAE is $$\bar{h} = \frac{\Delta h}{\Delta h_{\mathrm{train}}}, \qquad \bar{t} = \frac{t}{T}, \qquad \bar{v}_0 = \frac{v_0}{v_{0,\max}},$$2 m at $$\bar{h} = \frac{\Delta h}{\Delta h_{\mathrm{train}}}, \qquad \bar{t} = \frac{t}{T}, \qquad \bar{v}_0 = \frac{v_0}{v_{0,\max}},$$3 s, $$\bar{h} = \frac{\Delta h}{\Delta h_{\mathrm{train}}}, \qquad \bar{t} = \frac{t}{T}, \qquad \bar{v}_0 = \frac{v_0}{v_{0,\max}},$$4 m at $$\bar{h} = \frac{\Delta h}{\Delta h_{\mathrm{train}}}, \qquad \bar{t} = \frac{t}{T}, \qquad \bar{v}_0 = \frac{v_0}{v_{0,\max}},$$5 s, and $$\bar{h} = \frac{\Delta h}{\Delta h_{\mathrm{train}}}, \qquad \bar{t} = \frac{t}{T}, \qquad \bar{v}_0 = \frac{v_0}{v_{0,\max}},$$6 m at $$\bar{h} = \frac{\Delta h}{\Delta h_{\mathrm{train}}}, \qquad \bar{t} = \frac{t}{T}, \qquad \bar{v}_0 = \frac{v_0}{v_{0,\max}},$$7 s. The corresponding velocity MAEs are $$\bar{h} = \frac{\Delta h}{\Delta h_{\mathrm{train}}}, \qquad \bar{t} = \frac{t}{T}, \qquad \bar{v}_0 = \frac{v_0}{v_{0,\max}},$$8 m/s, $$\bar{h} = \frac{\Delta h}{\Delta h_{\mathrm{train}}}, \qquad \bar{t} = \frac{t}{T}, \qquad \bar{v}_0 = \frac{v_0}{v_{0,\max}},$$9 m/s, and $\Delta h_{\mathrm{train}}$0 m/s. The reported interpretation is that accuracy remains consistently high across time steps ranging from $\Delta h_{\mathrm{train}}$1 to $\Delta h_{\mathrm{train}}$2 s, with water-level errors on the order of $\Delta h_{\mathrm{train}}$3 m and velocity errors on the order of $\Delta h_{\mathrm{train}}$4 m/s (Shin et al., 3 Apr 2026).

The framework also generalizes to five distinct unseen initial conditions without retraining. At $\Delta h_{\mathrm{train}}$5 s, the reported water level MAE ranges from about $\Delta h_{\mathrm{train}}$6 to $\Delta h_{\mathrm{train}}$7 m, while the velocity MAE ranges from about $\Delta h_{\mathrm{train}}$8 to $\Delta h_{\mathrm{train}}$9 m/s. Within the scope of the trained parameter range, this indicates that one trained network can be reused across multiple initial conditions rather than retrained case by case (Shin et al., 3 Apr 2026).

The computational picture is more mixed. Relative to conventional iterative FDM solvers, P2F replaces the nonlinear momentum equation by a single network forward pass, introduces no retraining for new initial conditions within the training range, and remains compatible with existing node-based system-code structure. However, in the simplified verification case the hybrid P2F implementation is actually slower than the reference FDM solver, with about $T$0 speedup ratio, or roughly $T$1 slower. The explanation given is that the simplified FDM problem is already relatively cheap, so neural inference and data-transfer overhead dominate. The paper therefore presents the current result not as a universal runtime advantage, but as a coupling methodology whose benefit may emerge more clearly in more complex nonlinear and strongly coupled settings (Shin et al., 3 Apr 2026).

6. Relation to adjacent PINN–solver paradigms

P2F belongs to a broader trend in which governing parameters are embedded directly into PINNs so that a single network represents a family of solutions. In one-dimensional acoustics, a parameterized PINN was constructed by treating source location $T$2 as an input and learning $T$3 under wave-equation, initial-condition, and impedance-boundary constraints. In that formulation, the network learns a family of solutions rather than one solution per source location, and the authors explicitly note that this is closely aligned with the broader idea of parameterized PINNs and coupled physics systems (Borrel-Jensen et al., 2021).

A second nearby line of work replaces automatic differentiation in the PINN loss with finite-difference stencils on a grid. For Laplace’s equation and Burgers’ equation, the FDM-PINN approach computes the physics loss from a discrete residual $T$4 rather than from an AD-derived continuous operator, making the training resemble a numerical PDE solver operating on network predictions. This method is described as faster to compute than AD-based PINNs and on par in terms of error reduction in the reported experiments. That architecture is related to P2F in spirit but not identical in mechanism: FDM-PINN modifies how the residual is evaluated inside the neural loss, whereas P2F uses a parameterized PINN as a surrogate for one equation and couples it to an external FDM solver for another equation (Lim et al., 25 Feb 2026).

Parameterized PINNs for incompressible flow provide a further point of comparison. In the lid-driven cavity problem, Reynolds number was treated as an explicit input so that a single network represented $T$5. That work showed that pure parameterized PINNs are effective at low $T$6, but that hybridization with transfer learning and sparse CFD supervision becomes necessary as convection dominance increases and optimization imbalance worsens. This comparison is relevant because it illustrates a general limitation of pure parameterized PINNs: the parameterized manifold can be learned robustly only within regimes where training remains stable (Jangir et al., 4 Feb 2026).

These comparisons clarify a recurring misconception. “Parameterized PINNs coupled with FDM” does not simply mean any PINN that uses finite differences somewhere in its workflow. In the specific sense established by P2F, the defining features are a parameterized, data-free surrogate for a selected subsystem, solver-level coupling in a time-marching loop, and exact discrete mass conservation supplied by the FDM component (Shin et al., 3 Apr 2026).

7. Limitations, assumptions, and prospective extensions

The current P2F demonstration is subject to explicit assumptions. The verification problem is an idealized open-tank system, flow direction is fixed and sequential, pressure coupling between control volumes is negligible, and the momentum equation is simplified enough to allow node-wise sequential evaluation. The scope is currently limited to the CVH/FP module and has not yet been extended to MELCOR’s other major modules such as Heat Structure (HS) and Radionuclide (RN) (Shin et al., 3 Apr 2026).

The paper identifies several directions for future work: extension to closed systems with non-negligible pressure differences, bidirectional flow, the full matrix-based implicit momentum coupling used in MELCOR, integration with additional MELCOR packages for full multi-physics simulation, and performance studies under more complex regimes where the hybrid approach may become cost-competitive. These are stated research directions rather than established capabilities (Shin et al., 3 Apr 2026).

A broader implication is suggested by adjacent work. Parameterization by source position in acoustics and by Reynolds number in incompressible flow both indicate that a single PINN can learn a continuous solution manifold across a controlled parameter range (Borrel-Jensen et al., 2021, Jangir et al., 4 Feb 2026). Coupling that parametric surrogate to a discrete conservation update, as P2F does for thermal hydraulics, suggests a general strategy for problems in which one subset of equations is expensive and nonlinear while another must retain exact discrete conservation. This suggests a family resemblance among parameterized PINNs, hybrid PINN–solver frameworks, and classical reduced-order or surrogate-assisted simulation, while leaving open the question of when the hybrid overhead is justified by system-level complexity rather than by isolated benchmark cost.

Within its stated scope, P2F is therefore best understood as a numerical coupling methodology: a node-assigned hybrid framework in which a parameterized, data-free PINN learns the momentum response manifold over $T$7, and an FDM solver advances the conserved state. Its significance lies less in replacing conventional solvers outright than in specifying a reusable, conservation-aware interface between parameterized PINN inference and time-marching thermal-hydraulic simulation (Shin et al., 3 Apr 2026).