A Data-Free Physics-Informed Neural Operator for Level-Set Interface Advection

Muhammad Akbar Khan

cs.LGSep 18, 2026

A Data-Free Physics-Informed Neural Operator for Level-Set Interface Advection

Authors: Muhammad Akbar Khan

Organizations: Department of Mathematics, NED University of Engineering and Technology, Karachi 75270, Pakistan

Abstract

Operators for interfacial problems are trained on reference solutions produced by the solver they are intended to replace. This work develops a data-free physics-informed neural operator for level-set interface advection, in which the interface is the equation's unknown and the operator maps an initial interface to the full spatiotemporal trajectory under a prescribed flow. Training uses only the transport residual and a geometric constraint; no reference solution enters the objective at any point. A spacetime Fourier backbone emits the entire trajectory in one pass, and the initial condition is imposed by construction rather than by penalty, which removes the competition between the anchoring term and the residual that otherwise arises when no solution data are available. Supervised and hybrid operators are trained under an identical architecture, family, budget and test set, and are reported throughout as baselines that quantify what refusing labels costs. On a reversed single vortex the data-free operator reaches 1.614 +/- 0.067% relative L2 error on 100 held-out initial interfaces against 0.369 +/- 0.035% for the supervised baseline, a factor of 4.4; on solid-body rotation the corresponding figures are 3.804 +/- 1.075% and 2.576 +/- 0.159%, a factor of 1.5. The two benchmarks rank the arms differently, and the difference is attributable to the eikonal constraint: the exact solution violates |grad phi| = 1 over 0.3% of the domain under rotation and 86.9% under the vortex. Where the constraint is valid the physics-trained operator conserves enclosed area 2.7 times better than the supervised baseline despite a larger field error, and a hybrid arm using eight reference solutions outperforms a supervised arm using sixteen.

Figures & tables

Figure 1: The operator maps an initial interface to the full spatiotemporal trajectory under a prescribed flow. Its parameters are determined from the transport residual and the eikonal term alone; no reference solution enters training.

	median	fraction $>10^{-3}$
RO	$2.4\times 10^{-8}$	$0.3\%$
RV	$8.8\times 10^{-2}$	$86.9\%$

Table 1: Eikonal residual

(|\nabla\phi|-1)^{2}

of the exact solution, evaluated on the

64\times 64\times 32

grid.

	Parameter	Range
RO	orbit radius $\rho$	$[0.18,\ 0.26]$
	phase $\alpha$	$[0,\ 2\pi)$
	circle radius $R$	$[0.10,\ 0.18]$
RV	centre $c_{x}$	$[0.40,\ 0.60]$
	centre $c_{y}$	$[0.68,\ 0.82]$
	circle radius $R$	$[0.10,\ 0.18]$

Table 2: Family parameters. For RO the circle of radius

R

is centred at distance

\rho

from

(\tfrac{1}{2},\tfrac{1}{2})

at phase

\alpha

; for RV it is centred at

(c_{x},c_{y})

Arm	Labels	Relative $L^{2}$ (%)	Area error (%)	Drift (%)
Data-free	$0$	$3.804\pm 1.075$	$\mathbf{3.15\pm 0.49}$	$3.28\pm 0.47$
Hybrid	$8$	$\mathbf{1.599\pm 0.084}$	$3.42\pm 0.28$	$3.54\pm 0.29$
Supervised	$16$	$2.576\pm 0.159$	$8.61\pm 0.97$	$8.91\pm 0.99$
Exact solution	—	$0$	$0$	$0.82$

Table 3: Solid-body rotation, varying initial condition. Sixteen training instances,

30{,}000

iterations, three seeds,

100

held-out instances. Area error is measured against the reference field ( 15 ) and drift against the prediction’s own initial area ( 16 ); the latter requires no reference and its discretisation floor is given in the last row.

Figure 2: Solid-body rotation: predicted zero level set (solid) against the exact interface (dashed) at five times, for one held-out initial interface. Rows are the three arms at seed

42

Figure 3: Solid-body rotation: relative

L^{2}

error on each of the

100

held-out instances, sorted (left) and as a distribution (right), seed

42

Figure 4: Solid-body rotation at the final time: exact field, prediction and signed error for the same held-out instance as Figure 2 . The dashed contour marks the exact interface. Note the differing colour scales.

Arm	Labels	Relative $L^{2}$ (%)	Area error (%)	Drift (%)
Data-free	$0$	$1.614\pm 0.067$	$3.57\pm 0.79$	$3.68\pm 0.78$
Hybrid	$8$	$0.986\pm 0.081$	$2.73\pm 0.56$	$2.83\pm 0.60$
Supervised	$16$	$\mathbf{0.369\pm 0.035}$	$\mathbf{1.13\pm 0.10}$	$\mathbf{1.50\pm 0.13}$
Exact solution	—	$0$	$0$	$1.09$

Table 4: Reversed vortex, varying initial condition. Sixteen training instances,

20{,}000

iterations, three seeds,

100

held-out instances. Columns as in Table 3 .

Figure 5: Reversed vortex: predicted zero level set (solid) against the exact interface (dashed) at five times, for one held-out initial interface. Rows are the three arms at seed

42

Figure 6: Reversed vortex: relative

L^{2}

error on each of the

100

held-out instances, sorted (left) and as a distribution (right), seed

42

Figure 7: Reversed vortex at the final time: exact field, prediction and signed error for the same held-out instance as Figure 5 . The dashed contour marks the exact interface. Note the differing colour scales.

	Configuration	Relative $L^{2}$ (%)	Area error (%)	$w_{\text{eik}}$
RO	zero seeding (as used)	$4.042$	$\mathbf{3.15}$	$1.3\times 10^{-3}$
	ratio seeding (published)	$\mathbf{3.417}$	$3.99$	$2.7\times 10^{-3}$
	soft initial condition	$6.019$	$11.50$	$4.1\times 10^{-3}$
	no eikonal term	$18.968$	$5.76$	—
RV	zero seeding (as used)	$1.543$	$\mathbf{2.67}$	$6.3\times 10^{-5}$
	ratio seeding (published)	$24.812$	$73.64$	$3.7\times 10^{-3}$

Table 5: Data-free arm, seed

42

, with the eikonal treatment and the initial condition varied one at a time. Sixteen training instances,

100

held-out instances;

30{,}000

iterations for rotation and

20{,}000

for the vortex. The final column is the eikonal weight at the end of training.

Figure 8: Eikonal weight against training iteration for the three configurations of Table 5 that carry the eikonal term, seed

42

. Note the logarithmic ordinate. The soft initial condition holds the weight within a factor of two of its clamp for the whole of the vortex run.

	Relative $L^{2}$ (%)		Area error (%)
Arm	$20{,}000$	$30{,}000$	$20{,}000$	$30{,}000$
Data-free	$4.716\pm 1.277$	$3.804\pm 1.075$	$4.77\pm 0.74$	$3.15\pm 0.49$
Hybrid	$1.792\pm 0.119$	$1.599\pm 0.084$	$3.84\pm 0.40$	$3.42\pm 0.28$
Supervised	$2.621\pm 0.185$	$2.576\pm 0.159$	$9.13\pm 1.30$	$8.61\pm 0.97$

Table 6: Solid-body rotation at two optimisation budgets, all other settings identical. Three seeds,

100

held-out instances.

PDE Operator Learning

Partial differential equation (PDE) operator learning builds models that map inputs such as coefficients and initial conditions to solution fields.

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Eikonal Regularisation in Physics-Informed Neural Networks for Three-Dimensional Level-Set Advection: Transferability of Two-Dimensional Design Principles

Muhammad Akbar Khan

Department of Mathematics, NED University of Engineering and Technology, Karachi 75270, Pakistan

Neural PDE Solvers Physics-Informed ML Computational Fluid Dynamics

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Physics-informed neural networks applied to the level-set formulation of interface advection commonly augment the residual and initial-condition losses with an eikonal regulariser, penalising the deviation of

\|\nablaφ\|

from unity. A previous two-dimensional study identified this weight as the dominant hyperparameter and found its optimum shifts by four orders of magnitude between rigid-body and deforming flows, but left open whether these principles transfer to three dimensions and whether single-seed results survive run-to-run variability. We answer both by repeating the weight selection across four 3D benchmarks (translating sphere, rotating sphere, slotted sphere, reversed vortex), sweeping six weights with three seeds at full training budget under a pre-registered selection rule. The ordering transfers: the selected weight tracks how far the exact solution departs from the signed-distance property, spanning four decades from

10^{-1}

where it holds exactly to

10^{-5}

where the interface is stretched. Values transfer only benchmark by benchmark; two of four carry over unchanged and two do not, so inheritance must be verified. The multi-seed protocol reveals that at small weights the seed-to-seed standard deviation equals the error itself, and the regulariser reduces it by more than an order of magnitude, buying reproducibility as well as accuracy. We benchmark against a fifth-order WENO solver on identical grids and error measures; the classical scheme is more accurate on all four problems, by two orders of magnitude on smooth rigid advection, with a margin that narrows with geometric difficulty and is smaller in volume conservation than in the field norm. Finally, we show that the relative

L_2

error cannot certify the preservation of thin features, and report a feature-restricted measure that can.

Abstract

PDE Operator Learning

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