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Deep-Learning Solvers and Surrogates for Infinity and p-Laplace Problems

Tak Shing Au Yeung, Ka Chun Cheung, Hannah Potgieter, Steven J. Ruuth, Simon See

arXiv:2609.30809v1math.NAcs.LG

TL;DR

Large-p p-Laplace and infinity-Laplace problems pose computational challenges, especially in three dimensions. The paper studies PINNs and DeepONets for these problems, combining conditional convergence analysis with numerical experiments. The experiments report accurate recovery of infinity-Laplace solutions and limiting p-Laplace behavior, while the convergence results remain conditional on assumptions not guaranteed by the numerical optimization.

  • Problem

    Large-p p-Laplace problems can be ill-conditioned, and mesh-based solvers become costly in three dimensions; existing PINN convergence results generally do not cover these degenerate elliptic problems.

    Source: §1 · pp. 1–2

  • Method

    The paper uses PINNs for fixed PDE instances with stabilization and p-continuation, and DeepONets as data-trained surrogates across parameters and geometries.

    Source: §1 · pp. 2–3, §7 · p. 37

  • Results

    PINNs and DeepONet recover infinity-Laplace solutions and capture limiting p-Laplace behavior, with errors comparable to FEM convergence-to-limit baselines across finite-p experiments.

    Source: §7 · p. 37

  • Takeaways & Limitations

    A single DeepONet can represent solutions across p values and domain geometries, and including limiting-problem information can improve large-p interpolation and extrapolation.

    Source: §7 · p. 37

    PINN convergence results are conditional on residual and boundary errors vanishing in suitable norms, which finite-point training does not explicitly guarantee.

    Source: §4.3 · p. 9

Abstract

from arXiv · show

We investigate the use of neural network solvers for infinity and $p$-Laplace problems, which are fundamental in nonlinear analysis and have practical applications. Our approach employs Physics-Informed Neural Networks (PINNs) and Deep Operator Networks (DeepONets) to address computational challenges associated with large $p$ values, ranging from $2$ to $1000$, on various 2D and 3D domains. Our method offers advantages over traditional physics-based solvers, especially in three dimensions where mesh-based solvers become very costly for these problems. We also establish conditional convergence results for PINN approximations of both problems and a universal approximation result for DeepONet on the parametric $p$-Poisson problem. We demonstrate the effectiveness of these neural network solvers through numerical experiments and compare their performance with conventional methods.

1 Introduction

The paper studies neural-network solvers and surrogates for infinity- and p-Laplace problems, whose numerical treatment is challenging at large p and in three dimensions. It combines PINN convergence analysis and stabilization strategies with DeepONet operator learning, and evaluates both against conventional methods.

1 Introduction

  • Large p can make p-Laplace systems ill-conditioned, while the infinity Laplacian lacks divergence structure and is expensive to compute accurately for nonsmooth data.Traditional methods such as Oberman's become very expensive beyond two dimensions.Source: §1 · p. 1
  • Existing PINN convergence theory largely excludes degenerate elliptic operators such as the p-Laplacian and infinity Laplacian, which this work addresses under stated assumptions.The paper also assesses accuracy and robustness empirically against reference solutions.Source: §1 · p. 2
  • The paper adopts continuation-based training ideas for stabilizing large-p optimization, while explicitly withholding theoretical convergence guarantees for that setting.The approach is inspired by penalty and augmented-Lagrangian strategies that increase penalty parameters iteratively and reuse each solution as the next initial guess.Source: §1 · p. 2
  • DeepONet is used as a surrogate trained on conventional-solver or PINN data to interpolate p-Laplace solutions across parameter values, rather than as a standalone PDE solver.The study investigates generalization for large p and the p →∞ limiting behavior.Source: §1 · p. 2
  • The work presents a systematic PINN and DeepONet study for infinity- and p-Laplace problems over p = 2 to 1000, with conditional PINN convergence results and large-p training strategies.The strategies include η-normalization or regularization, residual clipping, top-2% outlier removal, gradient-norm clamping, and continuation in p.Source: §1 · pp. 2–3

2 Model problems and boundary value formulations

The section introduces p-Laplace and infinity-Laplace problems, including mixed boundary conditions and distance-approximation settings. It motivates large-p formulations and describes the parametric problem targeted by DeepONet.

2 Model problems and boundary value formulations

  • The infinity Laplacian arises formally as p →∞ for the homogeneous p-Laplacian, but lacks divergence form and is interpreted in the viscosity sense.It is a limiting equation only under specific assumptions on the source term and boundary conditions.Source: §2 · p. 3
  • Distance-function approximation is a target application, and nonsmooth examples such as |x| provide a test of solver robustness to limited regularity.The paper connects distance approximation to surface reconstruction, medical imaging, and computational mechanics.Source: §2 · p. 3
  • The mixed p-Laplacian boundary problem uses Γ1 as the feature set for geodesic distance and Γ2 as the remaining unconstrained boundary portion.Figure 1 depicts Γ1 with Dirichlet conditions and Γ2 with Neumann conditions.Source: §2 · p. 4, Figure 1 · p. 4
  • Large p values are difficult for traditional solvers because of sensitivity to initial guesses and ill-conditioning, motivating neural-network solvers and surrogates.The paper evaluates them empirically and gives PINN convergence analysis under structural assumptions not directly enforced in implementation.Source: §2 · p. 4
  • For the parametric p-Poisson problem, DeepONet can approximate the solution operator on compact parameter sets under a continuity assumption on the solution map.The study uses Newton-based FEM solutions as DeepONet training data and comparison baselines, and evaluates behavior across p values and geometries.Source: §2 · pp. 4–5

3 Data generation

The section constructs finite-element reference solutions for finite-p and limiting problems, then uses those solutions as comparison baselines and DeepONet training data. A damped Newton method with regularization and continuation supports the computations, although the FEM continuation typically reaches only about p = 200.

3 Data generation

  • A p-Laplacian finite-element solver generates reference solutions for finite-p and limiting infinity-Laplace cases, which serve as PINN baselines and DeepONet training data.The p-Laplacian's divergence structure supports the variational finite-element approach; FEM is also described as advantageous for nonsmooth problems.Source: §3 · pp. 5, 7
  • The nonlinear p-Poisson problem is solved with a damped Newton iteration on quadrilateral 2D or hexahedral 3D meshes using first-order elements.First-order elements are used because p-Laplace solutions are generally not C2 for p > 2.Source: §3 · p. 5
  • Regularization makes the generalized stiffness matrix positive definite, enabling conjugate gradients; an SSOR preconditioner with ω = 1.2 accelerates the solve.The Newton update uses quadratic-interpolation line search damping with a sufficient-decrease condition.Source: §3 · p. 6
  • Continuation in p reuses each converged solution as the next initial guess, and continuation with line-search damping improves robustness, particularly at large p.The sequence starts at p0 = 2, and the next solve is initialized from the previous p solution.Source: §3 · p. 7
  • The FEM continuation typically reaches approximately p = 200 before failing to converge, often because no suitable descent direction is found.The experiments use p0 = 2, an increment of 1, and tolerance 10^-8.Source: §3 · p. 7

4 Physics-Informed Neural Networks

This section presents PINNs for fixed p-Laplace and infinity-Laplace problems and their conditional convergence theory. It distinguishes continuum convergence under stated structural assumptions from guarantees about numerical training, since those assumptions are not directly enforced by the implemented loss.

4.1 Network architecture

  • The PINN uses smooth tanh activations, four-layer blocks with 128 neurons, and a final single-output layer to approximate potentially nonsmooth solutions.Smooth network outputs are used despite the possibility that target solutions have limited regularity.Source: §4.1 · p. 7

4.2 Collocation sets and loss terms

  • Training combines boundary-condition losses at boundary collocation points with an interior PDE-residual loss, using the relevant p-Laplace or infinity-Laplace operator.Boundary derivatives are computed by automatic differentiation, and Adam minimizes the total loss with a balancing weight α.Source: §4.2 · pp. 8–9
  • To stabilize large-p training, the method clamps |∇u| to at most 1 in the PDE loss and raises α from 10^-5 to at most 10^-1 when boundary loss plateaus.This staged weighting prioritizes boundary satisfaction before progressively enforcing the PDE residual.Source: §4.2 · p. 9

4.3 PINNs convergence under structural assumptions

  • For p-Laplace problems, PINN sequences converge weakly in W^1,p under the theorem’s assumptions, and strongly when boundary conditions are enforced exactly.The result concerns continuum residual and boundary-error assumptions, not convergence guarantees for the discrete optimization used in the experiments.Source: §4.3 · pp. 9, 14, 15
  • For the infinity-Laplace problem, vanishing residual and boundary errors plus uniformly Lipschitz PINN approximations imply uniform convergence to the viscosity solution.The theorem’s assumptions idealize successful training and do not guarantee convergence of the finite-collocation optimization procedure.Source: §4.3 · pp. 9, 15, 16
  • Uniform boundedness and equicontinuity yield a uniformly convergent subsequence on ¯Ω, providing the compactness step in the convergence proof.The argument uses the Arzelà–Ascoli theorem.Source: §4.3 · p. 16
  • The limit lies in W 1,∞(Ω) ∩ C(¯Ω), satisfies the boundary condition u = g, and is both a viscosity subsolution and supersolution.The proof verifies the viscosity conditions using test functions touching the limit from above and below.Source: §4.3 · pp. 17–18
  • The entire PINN sequence converges uniformly to the unique viscosity solution u* of the Dirichlet problem for -∆∞u = 0.Every subsequence has a further subsequence converging uniformly to the same limit; uniqueness follows from Jensen’s theorem [3].Source: §4.3 · p. 18

5 DeepONet

DeepONet serves as a surrogate for parametric p-Laplace problems, learning solution fields across p values and, where relevant, domain geometry. The section describes its branch–trunk architecture and training data, then establishes a conditional universal approximation result for the parametric p-Poisson solution operator.

5 DeepONet

  • DeepONet learns a mapping from problem inputs to solution fields, allowing one trained model to predict across p values and, where relevant, domain geometries.It is a surrogate trained on high-fidelity numerical data, not a direct PDE solver.Source: §5 · p. 18
  • The approximation result does not provide quantitative error estimates or rigorous convergence guarantees for degenerate elliptic operators such as the p-Laplacian and infinity Laplacian.The models are therefore evaluated empirically against reference solutions and for their behavior at large p.Source: §5 · p. 19

5.1 DeepONet architecture

  • The trunk network encodes spatial coordinates, while the branch network encodes p and, when relevant, geometric parameters; their outputs are combined to produce the solution.In the experiments, each network has three hidden fully connected layers; the trunk uses 512 neurons per layer and the branch 128, with latent dimension 128 and tanh activations.Source: §5.1 · p. 19

5.2 Training data and loss function

  • Training uses reference solutions from the Newton-based finite element solver for selected p values and PINN approximations for the infinity-Laplacian limit.Training across parameter instances lets the model learn dependence on p and domain geometry and predict at unseen p values.Source: §5.2 · p. 20

5.3 Universal approximation for p-Laplace Problems

  • For the parametric p-Poisson problem, DeepONet can uniformly approximate the solution mapping on compact parameter sets if the solution operator is continuous.The proposition assumes bounded domains with C1,β boundaries, 2 ≤ p ≤ P < ∞, and unique weak solutions that are continuous; the approximation uses branch input z = (µ, p) and trunk input x.Source: §5.3 · p. 20, Define the pullback · pp. 20–21

Define the pullback

  • The proof represents each solution on a fixed reference domain and treats the resulting pullback as a continuous operator from parameters to functions.The parameter includes µ and p, and the theorem assumes continuity of this solution operator.Source: §5.3 · p. 20, Define the pullback · pp. 20–21

6 Numerical experiments

The numerical experiments assess PINNs and DeepONets for infinity- and p-Laplace problems, focusing on large p, reduced regularity, and varying domain geometries. They examine accuracy against fixed-resolution FEM baselines and limiting distance profiles, as well as challenges in direct finite-p prediction and the distinction between interpolation and extrapolation.

6.1 Testing error

  • Reported errors generally compare learned predictions with u∞, measuring recovery of the limiting profile; the unit-disc validation also reports direct error against the exact finite-p solution.The 2D PINN test set uses a uniform 100 × 100 grid restricted to the domain, together with boundary points.Source: §6 · p. 21, §6.1 · pp. 22–23

6.2 2D infinity Laplace equation with Dirichlet boundary conditions

  • The η-normalized infinity-Laplacian residual removes the bound’s dependence on the prediction’s gradient norm, improving training stability.The normalization uses η = 10−5 and preserves the same solution set; residual clipping and outlier removal further address optimization-level robustness.Source: §6.2 · p. 24

6.2.1 Arctan example (homogeneous, Dirichlet)

  • For the Arctan problem, PINN test MSE was (3.118 ± 2.327)×10−7 versus FEM error 5.892 × 10−5, more than two orders of magnitude lower.PINN inference took 1.456 × 10−4s over the domain, compared with approximately 2776s for FEM computation.Source: §6.2.1 · p. 25

6.2.2 Aronsson example (homogeneous, Dirichlet)

  • For the Aronsson square, PINN test MSE was (4.468±1.696)×10−7, more than two orders of magnitude below the fixed-resolution FEM error of 9.511 × 10−5.The five-seed mean and sample standard deviation quantify the PINN test error.Source: §6.2.2 · pp. 25–26
  • On the Aronsson unit disc, PINN test MSE was (6.247 ± 2.805) × 10−7, more than two orders of magnitude below the fixed-resolution FEM error of 2.522 × 10−4.PINN inference took 7.608 × 10−5s, whereas the Newton-based FEM solver required approximately 17447s.Source: §6.2.2 · p. 26

6.3 p-Laplace equation with Dirichlet boundary conditions

  • Independently trained p-Laplace PINNs do not show monotonically decreasing MSE∞ as p increases and may become unstable at large p.The reported explanation is that p-dependent residual nonlinearities can cause numerical instability and loss imbalance when initialization is far from the true solution.Source: §6.3.1 · p. 28
  • With iterative continuation from p = 2 to p = 1000, MSE∞ initially decreases as p rises.The procedure uses five continuation bands, fine-tuning checkpoints from the preceding band for successive p values.Source: §6.3.2 · p. 28
  • Iterative PINN training stabilizes boundary loss and limiting-solution error as p increases, enabling continuation to p = 1000, while simple training eventually fails at large p.The p = 1000 entries are continuation diagnostics, not fully verified finite-p solves; the final continuation band reduces the PDE residual weight to α = 10−5.Source: Table 5 · p. 29, §6.3.2 · p. 29

6.3.3 Finite-p validation on the unit disc

  • MSEp remains below 3 × 10−5 at every reported p, reaching 1.922 × 10−6 at p = 1000.MSE∞ decreases from 7.145 × 10−2 at p = 2 to 2.385 × 10−6 at p = 1000, tracking convergence toward the distance-to-boundary limit.Source: §6.3.3 · p. 30, Table 6 · p. 31

6.4 DeepONet example

  • DeepONet is trained on finite-p solution data to recover the p →∞ limiting solution, and its MSE∞,p measures limiting-solution error rather than finite-p solution error.The exception is direct unit-disc finite-p validation against the exact solution, reported separately in Table 9.Source: §6.4 · p. 31

6.4.1 Distance-to-origin DeepONet example

  • Adding the PINN-predicted p = ∞ solution as a p = 500 training input controls MSE∞,p at larger p; predictions for 200 < p < 500 are interpolation, not extrapolation.Without the limiting-solution input, MSE∞,p continues increasing beyond p = 200, the upper end of the finite-p training data.Source: §6.4.1 · p. 32, Table 7 · p. 33, Figure 8 · p. 34

6.4.2 Distance-to-boundary DeepONet 2D and 3D examples

  • DeepONet recovers limiting distance profiles with errors comparable to the FEM convergence-to-limit baseline on 2D domains and remains effective on more challenging 3D domains.Training takes approximately 17,500 s, while Newton FEM computations in 3D require up to approximately five days; once trained, DeepONet evaluates multiple p values and geometries without retraining.Source: §6.4.2 · p. 33, Table 8 · p. 34
  • For 3D domains, MSE∞,p trends vary: behavior is most consistent on the sphere, while the torus error remains above the FEM baseline and ellipse-2 errors can rise beyond the FEM training range.On the unit disc, direct DeepONet error MSEp remains below 1.6 × 10−6 across p, including unseen interpolation points, against exact finite-p solutions.Source: §6.4.2 · pp. 33–34, Figure 10 · p. 36, Table 9 · p. 37

6.4.3 Distance-to-boundary DeepONet over family of ellipses

  • On previously unseen ellipse geometries, DeepONet achieves MSE(θ) generally of order 10−4 at p = 500 across the rotation range.The experiment tests interpolation over ellipse parameters (θ, a, b) not included in training.Source: §6.4.3 · p. 36
  • With too few training parameter samples, DeepONet overfits and predicts poorly at intermediate rotations; increasing sample count improves approximation and smooths the error profile.The parameter-sampling experiment varies the number of evenly spaced θ values while holding the remaining parameters fixed.Source: §6.4.3 · pp. 36–37

7 Conclusions

The experiments show that PINNs and DeepONet recover infinity-Laplace solutions and p-Laplace limiting profiles, with PINN stabilization improving as p grows. DeepONet represents solutions across p and domain geometries in one trained operator, and the conclusion identifies further application areas.

7 Conclusions

  • PINNs and DeepONet recover infinity-Laplace solutions and p-Laplace limiting profiles with errors comparable to FEM convergence-to-limit baselines across a wider p range.For PINNs, iterative training in p combined with u-gradient norm clipping improves stability and robustness as p increases.Source: §7 · p. 37
  • A single DeepONet can incorporate p and domain geometry, while information from the p →∞ limit can improve large-p interpolation and extrapolation beyond finite-p training data.This is suited to evaluating p-Poisson distances across multiple p values and geometries.Source: §7 · p. 37
  • PINNs can be trained from governing equations and boundary conditions without precomputed interior solution data, which is useful when high-quality training data is expensive or infeasible to generate.Source: §7 · pp. 37–38
  • At p = 500, the ellipse DeepONet evaluation varies rotation angle and semi-axis lengths, testing unseen configurations within the sampled parameter range.Figure 12 compares MSE(θ) for models trained with 3, 5, 7, and 9 rotation-angle values.Source: Figure 11 · p. 38, Figure 12 · p. 38
  • The conclusion points to surface formulations for p-Poisson geodesic distances and applications including image processing, elastoplastic torsion, and non-Newtonian fluid models.The cited application areas involve nonlinear and degenerate elliptic operators that challenge classical solvers.Source: §7 · p. 38

Code and data availability

The code, data-generation and plotting scripts, and experiment configuration files are available in the project repository.

Code and data availability

  • The repository provides code, data-generation and plotting scripts, and experiment configuration files, with fixed random seeds for reproducing reported runs and figures.Source: Code and data availability · p. 38

Funding/Acknowledgments

The authors acknowledge support from Canada's Natural Sciences and Engineering Research Council.

Funding/Acknowledgments

  • The work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), grant RGPIN-2022-03302.Source: Funding/Acknowledgments · p. 39

A Training hyperparameters

The appendix lists training hyperparameters for the PINN and DeepONet experiments and describes how inference times were measured.

A Training hyperparameters

  • Tables 10 and 11 list training hyperparameters for PINN experiments in Sections 6.2–6.3 and DeepONet experiments in Sections 6.4.1–6.4.2, respectively.Table 11 is the DeepONet training-hyperparameter table.Source: A Training hyperparameters · p. 43, Table 11 · p. 43
  • Inference times use batched test-set forward passes averaged over 100 runs after 10 warm-ups, with CUDA synchronization and input transfer and postprocessing excluded.Source: A Training hyperparameters · p. 43

B Ablation study for infinity Laplacian PINNs

The infinity-Laplacian ablations identify outlier removal as the most consequential stabilization component. Loss balancing, scheduling, and network depth also affect performance, while width has limited influence.

Stabilization

  • Disabling top-2% outlier removal increases the Aronsson disc error by approximately 200×, the largest adverse effect among the tested stabilization changes.The baseline combines η-normalization, residual clipping to [−10, 10], and top-2% outlier removal.Source: B Ablation study for infinity… · p. 44

Loss balancing

  • Setting α = 0 substantially increases errors in all cases because boundary loss alone does not enforce the governing equation.ReLoBRaLo produces substantially larger errors for all three test configurations.Source: B Ablation study for infinity… · p. 44

LR schedule

Width and depth

  • Increasing network depth improves performance for smooth examples, but four- and five-layer networks perform similarly; width 32 performs similarly to widths 128 and 256 in most cases.Source: B Ablation study for infinity… · p. 44

C Ablation study for simple p-Laplacian PINNs

The simple p-Laplacian ablations test stabilization at p = 10 and p = 200. Gradient-norm clamping is most critical, with its removal causing divergence at p = 200.

Stabilization

D Iterative training schedule for the p-Laplacian

The iterative p-Laplacian experiments train in successive bands of p, carrying the best checkpoint forward. The unit-disc validation uses a separate continuation schedule and staged training configuration.

D Iterative training schedule for the p-Laplacian

  • Each band loads the preceding band’s best checkpoint; band 1 may stop when boundary validation loss is below 10^-4 and PDE residual loss below 10^-3.Bands 2–5 use fixed α values and a constant learning rate of 10^-5.Source: D Iterative training schedule for the… · p. 45
  • The five-band homogeneous p-Laplacian schedule uses early stopping in band 1 and fixed epochs per p value in bands 2–5.Source: Table 14 · p. 45
  • The unit-disc finite-p validation uses a separate schedule with 10,000 boundary points, split into 8,000 training and 2,000 validation points at stage 1; later stages use all 10,000.Its four-hidden-layer, 128-neuron network resamples interior collocation points each epoch and carries the preceding stage’s final checkpoint forward.Source: Table 15 · p. 45, D Iterative training schedule for the… · p. 45

E Eikonal PINNs training for DeepONet data generation

Eikonal PINNs generate distance-function training data for DeepONet, using the limiting solution when available and an Eikonal PINN otherwise. The training setup includes domain-specific zero boundary conditions and a soft positivity constraint.

E Eikonal PINNs training for DeepONet data generation

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