Sparse observations can still rebuild two-phase flow: hierarchically weighted physics-informed networks lead tuned regressors at 20 labels
Synopsis
With only 20 to 50 coordinate–state observations, physics-informed neural networks (PINNs, which write the governing equations and initial and boundary conditions into the loss) reach the highest mean primary-target accuracy on a heat equation and three heterogeneous two-phase-flow benchmarks, and the water-saturation advantage holds through 1000 labels. Hierarchical loss weighting (H-PINN) and its gradient-direction-aware extension (HO-PINN) activate physical constraints as higher-priority losses approach prescribed scales, reducing run-to-run variability in selected settings. The cost is fitting time substantially higher than tuned gradient boosting, support-vector regression and Gaussian-process regression.
Interpretation
When labels are scarce, writing known physics into the training objective lets a neural surrogate reconstruct solution fields from only tens of coordinate–state observations, including heterogeneous two-phase flow with moving saturation fronts. Earlier data-driven porous-media surrogates typically train on complete simulation realizations or sequences of fields, whereas here each label is one coordinate and state value, not a complete simulation. At 20 and 50 labels, at least one PINN variant has the highest mean primary-target accuracy in all 16 test–region–budget comparisons; across 20 to 1000 labels, a PINN leads all 36 water-saturation comparisons in the flow tests, and the six classical wins are confined to the heat equation.
Hierarchical weighting lets physical constraints take effect in priority order instead of competing with the data term at fixed weight from the start. A vanilla PINN uses constant weights on data, initial/boundary and equation residuals, while H-PINN and HO-PINN normalize their weights and activate each later objective only as the preceding error approaches a prescribed scale. HO-PINN further down-weights gradient directions already covered by earlier objectives, using them only to form scalar weights rather than replacing the final backward gradient; on the heat equation the median standard-deviation ratios of H-PINN and HO-PINN to vanilla are 0.19 and 0.16.
This label efficiency comes with substantial fitting cost, and the stability gain from weighting depends on the problem. Vanilla PINN is more reliable than the weighted variants in some extremely sparse flow conditions, and Gaussian-process regression is preferable on the sufficiently sampled smooth heat solution. At 20 labels, average GPU fitting times across the four tests range from 39 to 96 minutes for vanilla, 41 to 97 for H-PINN and 64 to 158 for HO-PINN, while classical fitting including tuning takes about 0.8 to 66 seconds on CPU.
Perspective
The conclusions apply where governing equations and initial and boundary conditions are known and each label is a single coordinate–state observation, covering a periodic heat equation and three heterogeneous two-phase-flow benchmarks with permeability contrasts of about 24 and approaching five orders of magnitude, including a seven-input parametric surrogate. The flow references use one spatial resolution and fixed permeability fields, and Test 4 holds out coordinates rather than entire parameter combinations.
The stability gain from weighting is problem dependent: vanilla PINN has smaller standard deviation in most Test 2 conditions, its forecasting score at 20 labels in Test 3 is better than both weighted variants, and the weighted variants show a few negative primary values in Tests 3 and 4 at 20 labels. Development inspected predictive metrics and reused seeds 101–103, so the protocol is not fully nested model selection; whether the ranking holds when the equation is unavailable, uncertain or enforced only within the labelled interval is not established.