2026-09-03
A conservation nodal model with four 2×16 nets learns LN2 tank closures from 48 experiments: 2.5–2.8% out-of-fold global NRMSE, 1.6% in-sample, with sloshing the hardest regime.
Long-duration storage of cryogenic propellants sits in the middle of spaceflight and of hydrogen-fuel plans for aviation and shipping. LH2 lives near 20 K, LCH4 near 120 K. Heat and mass transfer inside the tank couple tightly, and sloshing makes the closures worse. CFD can resolve the physics. It is too expensive for long-horizon, system-level work, and it still needs empirical accommodation factors to match tests. Nodal models are cheap and conserve mass and energy, so they suit control and digital twins. They stall on the unresolved pieces: wall and interface heat transfer, phase change, pressurant quality, how the liquid thermal boundary layer grows, and how sloshing rips stratification apart. Reliable correlations are scarce in dynamic regimes, and even static cases often need extra fudge factors.
This line of work from the von Karman Institute embeds a neural net in a thermal nodal model and identifies the closures with adjoint optimization. An earlier paper already did that. The new pieces are experimental: 48 liquid-nitrogen runs covering self-pressurization, active pressurization, venting, and lateral sloshing; a first-order dynamical model for liquid thermal-boundary-layer thickness; and an entropy-production penalty in the loss so the closures are discouraged from violating the second law.
The tank is split into vapor, liquid, and solid control volumes. Conservation gives ODEs; the missing coefficients go to the networks. Four feed-forward nets, one per regime, each with two hidden layers of 16 SiLU units, output an 11-dimensional closure vector: eight Nusselt numbers, two boundary-layer parameters, and the pressurant vapor fraction χp. Positive quantities go through softplus; χp is sigmoid-mapped to [0, 1]. Inputs are dimensionless groups: Jakob and Rayleigh numbers, fill ratio, δT/R, a sloshing Reynolds number, plus a pressurant-rate group Πp for the active-pressurization and venting nets. Regimes switch with hard indicator functions. The authors flag fuzzy memberships as future work.
Training is adjoint-based. JAX and diffrax integrate the ODEs forward with Tsitouras's fifth-order Runge–Kutta, then integrate the adjoint backward. AdamW updates the weights at learning rate 10⁻³ and weight decay 10⁻⁴. Dropout is 0.1. Measurements get a 5% relative perturbation. The entropy penalty weight Kp=10⁵ was the smallest value, among 10² to 10⁶, that cut negative entropy production without hurting reconstruction. Pressure is weighted 2.0 in the loss; vapor and liquid temperature and vapor mass get 1.0; boundary-layer thickness gets 0.1 because the measurement is noisier.
The data come from VKI's CryME LN2 facility: a quartz tank with an 83 mm internal diameter and 124 mm internal height, residual pressure about 0.8 kPa, sixteen Lakeshore DT-670-SD diodes. Forty-eight multi-stage time series. Cross-validation is split at experiment level for K=2, 3, 4, 5. Held-out runs are integrated open-loop from the reconstructed initial state, with no intermediate resets. For sloshing, the measured interface amplitude As is an exogenous input. Predicting the free surface from tank motion is out of scope.
Training loss lands near (4.6±0.5)×10⁻³ for every K. Out-of-fold loss drops from 10.3×10⁻³ at K=2 to 7.3×10⁻³ at K=5, while the fold-to-fold standard deviation rises from 0.1×10⁻³ to 3.2×10⁻³. More training data helps the mean and makes the split matter more.
Global NRMSE averages the component-wise normalized RMS errors:
| Split | Overall | Self-press./relax. | Active press. | Venting | Sloshing |
| K=2 | 2.75±0.35% | 2.31% | 2.68% | 1.91% | 3.48% |
| K=4 | 2.53±0.49% | 2.31% | 1.89% | 0.70% | 3.62% |
| K=5 | 2.82±0.68% | 2.41% | 2.29% | 0.74% | 3.96% |
K=4 is the best overall. Venting can reach 0.7% once the training folds cover it. Sloshing stays the hardest; adding data does not monotonically help.
A final model trained on the full database is in-sample reconstruction, not generalization: 1.60% overall, 1.30% self-pressurization, 1.57% active pressurization, 0.65% venting, 2.30% sloshing. Pressure is 1.79% overall and 2.49% under sloshing. Thermal boundary-layer thickness is 10.30% overall and 16.13% under sloshing; relative error blows up when the layer collapses toward zero. Training loss settles at (6.58±0.01)×10⁻³.
This is a closure law for system-level control and digital twins, not a CFD replacement. The nets are tiny, the ODEs integrate over long horizons, and the outputs still have physical names: Nusselt numbers, quality, boundary-layer growth. For an online predictor on a cryogenic tank, a conservation skeleton plus small nets is more auditable than a black-box pressure regressor. The cross-validation says 48 information-rich LN2 runs can hold out-of-fold error under 3%, if geometry and fluid stay put.
Relative to the same group's prior paper, this is incremental: a broader experimental matrix, a stratification state, and an entropy term.
The 1.60% figure is reconstruction on the training set. The number that matters is the 2.5–2.8% out-of-fold band, and fold composition can swing that by ±0.7%. The fluid is LN2. The motivating applications often quote LH2 and LCH4, which live at different temperatures; the closures do not transfer by default. The tank is a small, ground-based upright cylinder. Geometry, scale, and gravity are single. Regime switching is hard, with no soft membership across transitions. The hardest regime still consumes measured wave amplitude; free-surface prediction is not in the model. The entropy term penalizes negative production and does not strictly enforce the second law. No architecture search. Optical fill level in violent three-dimensional sloshing needs an iterative correction from temperature and pressure, and wall-node temperatures rest on few sensors.