A wide gap on a singular source
A proposed physics-informed neural network (PINN) recorded much smaller average relative L2 errors than a strong-form L2 residual on a p-Laplace benchmark with a Dirac point source. Across the displayed exponent values, the weak-loss errors ranged from 2.8127 × 10^-3 to 4.8305 × 10^-3 when p > 2, and from 2.8461 × 10^-3 to 3.9188 × 10^-3 when p < 2. Strong-loss errors ranged from 9.9374 × 10^-1 to 1.0011 × 10^0 for p > 2 and were 1.0000 × 10^0 for p < 2. The reported comparison averages were calculated over five independent trials.
The result is part of a methods study asking whether a different way of training PINNs can remain mathematically controlled when the p-Laplace problem has low-regularity solutions, and whether the same design can handle equations whose exponent, source or boundary data vary. The document is identified as arXiv:2608.24205v1, dated 25 Aug 2026.
A loss built around weak solutions
The proposed loss measures the PDE mismatch with a dual residual norm and the boundary mismatch with a fractional Sobolev norm. The paper presents that pairing as a way to accommodate weak-solution regularity and to support rigorous a priori and a posteriori error estimates.
One consistency result states that the stated weak loss vanishes exactly at the unique weak solution. A separate theorem gives conditional a posteriori bounds for the W^{1,p} solution error in terms of the loss L(theta), with separate cases for p > 2 and 1 < p < 2. Those estimates are conditional, not unconditional training guarantees: the theorem assumes a near-convergence regime in which residuals, boundary errors and solution error are bounded by one.
For practical computation, the experiments use a truncated variational PINN, or VPINN, residual sum. Gaussian quadrature handles the spatial and weak-residual integrals, while Monte Carlo integration is used for empirical boundary and parameter losses. This is the practical approximation used for the numerical tests.
The pattern continued with regular data
The gap remained in a regular-source benchmark. At p = 5.00, the weak-loss average relative L2 error was 3.1378 × 10^-3, compared with 1.1629 × 10^0 for the strong loss. At p = 1.50, the corresponding figures were 1.9815 × 10^-4 and 3.7820 × 10^-1. The comparison averages were calculated over five independent trials.
A second comparison changed the way boundary mismatch was penalized. At p = 5.00, the fractional-boundary version had an average relative L2 error of 3.1050 × 10^-2, versus 9.9997 × 10^-1 with an L2 boundary loss. At p = 1.50, the figures were 4.0264 × 10^-4 and 4.3689 × 10^-4. The paper reports lower fractional-loss errors across nearly all tested p values and says the standard L2 boundary loss failed to train at p = 5, but one moderate-exponent row favored the L2 version slightly.
One model for changing equations
The study also embeds problem-defining parameters in the network input. Its parametric formulation is designed to let one network represent varying p-Laplace instances without retraining a separate model for each case. The reported experiments varied either the exponent or the forcing term and compared predictions with manufactured solutions.
For the exponent experiment, errors in the first parameter range, labelled P1, at mu = 2.05, 3.00, 4.00 and 5.00 were 1.2892 × 10^-3, 3.9298 × 10^-4, 7.3375 × 10^-4 and 9.9399 × 10^-4. In the second range, P2, at mu = 1.95, 1.80, 1.65 and 1.50, they were 1.5174 × 10^-3, 2.7256 × 10^-3, 3.8536 × 10^-3 and 7.0549 × 10^-3. The single model reproduced the manufactured solution across the tested exponent slices, according to the reported comparisons.
In the forcing experiment, the reported errors at p = 1.8 and mu = 1.5, 2.0, 2.5 and 3.0 were respectively 5.4303 × 10^-4, 8.2709 × 10^-4, 7.2281 × 10^-4 and 1.1338 × 10^-3. At p = 2.5, the corresponding values were 4.8773 × 10^-4, 3.8844 × 10^-4, 3.0850 × 10^-4 and 3.5475 × 10^-4. The paper describes these results as tracking the manufactured solution across the tested forcing terms.
What the numbers do not settle
The results have a narrower reach than the headline numbers might suggest. They come from synthetic p-Laplace benchmarks and selected exponent and parameter slices, and the parametric analysis does not report an independent held-out parameter evaluation. The theory is stated in W^{1,p} error, while the numerical endpoint is relative L2 error, so the supplied analysis does not quantify how closely those two measures track one another. The comparison averages were reported without trial-level dispersion or formal significance tests.
Training was also tied to a particular comparison setup: three hidden layers with 50 neurons per layer and tanh activation, followed by 20,000 Adam iterations and 5,000 L-BFGS iterations. That specificity limits what can be inferred about other architectures, optimizers or training regimes; the supplied results do not establish universal superiority or unconditional convergence.
Taken together, the paper's conditional theory and benchmark results support the authors' interpretation that the dual-residual, fractional-boundary design is robust for low-regularity nonlinear p-Laplace problems and that one parametric network can cover solution families without individual retraining. The acknowledgements report support from the National Research Foundation of Korea, the POSCO TJ Park Foundation, the Korea Institute for Advanced Study and the Yonsei University Research Fund.
Paper data and sources
Original title: Robust training and rigorous error analysis of physics-informed neural networks for the $p$-Laplace equation
Authors: Kyueon Choi, Seungchan Ko, Dohyun Kwon
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
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