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

math.NASep 25, 2026

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

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

Organizations: NVIDIA AI Technology Center, NVIDIA, Hong Kong, China · Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada · NVIDIA AI Technology Center, NVIDIA, Singapore, Singapore

Abstract

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.

Figures & tables

Figure 1: Example of a suitable 2D domain: $\Omega$ (grey) is an open subset of a rectangle , $\Gamma_{1}$ (orange) is the ‘feature’ set where the Dirichlet condition is imposed, and $\Gamma_{2}$ (black) is where the Neumann condition is imposed.

Figure 2: PINN architecture and training logic for the two dimensional setting. The three dimensional case is analogous.

Figure 3: DeepONet architecture used in this work.

Example	Equation of domain	Cell edge length			Mesh size
Example	Equation of domain	mean	max	min	# cells	# vertices	# faces
2D disc	$x^{2}+y^{2}\leq 1$	$2.5132$ e-02	$4.9083$ e-02	$1.8043$ e-02	$6825$	$5185$	$13776$
Ellipse 1	$x^{2}+16y^{2}\leq 1$	$1.1645$ e-02	$1.5783$ e-02	$6.4314$ e-03	$7644$	$6025$	$15638$
Ellipse 2	$8.5x^{2}+8.5y^{2}-15xy\leq 1$	$1.1645$ e-02	$1.5783$ e-02	$6.4314$ e-03	$7644$	$6025$	$15638$
Ellipse 3	$4x^{2}+y^{2}\leq 1$	$6.6203$ e-03	$1.8764$ e-02	$1.4471$ e-03	$46704$	$35785$	$93758$
3D ball	$x^{2}+y^{2}+z^{2}\leq 1$	$2.0717$ e-02	$4.9105$ e-02	$9.1529$ e-03	$224694$	$201857$	$680907$

Table 1: Summary of mesh statistics for training data used in the FEM scheme. All meshes are composed of quadrilateral or hexahedral elements and were generated with deal.II [ 1 ] . We report the average, maximum, and minimum cell edge lengths, as well as the total number of cells, vertices, and faces for each domain geometry.

Training:
$\mathrm{MSE}_{\partial\Omega}$	( 6 )	Enforces Dirichlet boundary data
$\mathrm{MSE}_{\partial\Omega,\mathrm{mixed}}$	( 7 )	Enforces mixed Dirichlet-Neumann boundary data
$\mathrm{MSE}_{\mathrm{PINNs}}$	( 8 )	Enforces governing equation
$\mathrm{MSE}$	( 9 )	Total training objective
Testing:
$\mathrm{MSE}_{\infty}$	( 26 )	Infinity Laplacian: error vs infinity solution

Table 2: Mean squared error quantities used for PINN training and evaluation.

Training:
$\mathrm{MSE}_{\mathrm{DeepONet}}$	( 24 )	Data-driven training objective
Testing:
$\mathrm{MSE}_{\infty,p}$	( 27 )	$p$ -Laplacian: error vs limiting infinity solution

Table 3: Mean squared error quantities used for DeepONet training and evaluation.

Example	Exact solution	$\\|u_{\infty}\\|_{2,N_{\mathrm{test}}}$	MSE		Runtime
			Our method	Newton	PINN inference	Newton
			(mean $\pm$ s.d.)	Newton	(+training)	Newton
Arctan, Fig 4	$\text{arctan}(y/x)$	$8.862$ e-01	$(3.118\pm 2.327)$ e-07	$5.892$ e-05	$1.456$ e-04s (+ $1453$ s)	$2776$ s
Aronsson (square), Fig 6	$\|x\|^{\frac{4}{3}}-\|y\|^{\frac{4}{3}}$	$4.277$ e-01	$(4.468\pm 1.696)$ e-07	$9.511$ e-05	$7.719$ e-05s (+ $1064$ s)	$6955$ s
Aronsson (disc), Fig 6	$\|x\|^{\frac{4}{3}}-\|y\|^{\frac{4}{3}}$	$4.284$ e-01	$(6.247\pm 2.805)$ e-07	$2.522$ e-04	$7.608$ e-05s (+ $905$ s)	$17447$ s

Table 4: Comparison of our PINN method and the Newton FEM method for solving the two-dimensional infinity-Laplacian equation subject to Dirichlet boundary conditions. The PINN MSE is reported as the mean

\pm

sample standard deviation over five runs with random seeds

1234

3456

5678

6789

, and

8765

. The reported PINN timings and loss histories in Figures 4 - 6 correspond to the seed-

1234

runs. The Newton FEM results are deterministic results computed on the fixed meshes described in Table 1 .

Panel of Figure 4 — Figure 4: Arctan example 6.2.1 . (a) Loss histories versus epoch: training boundary loss $\mathrm{MSE}_{\partial\Omega}$ , training PINNs loss $\mathrm{MSE}_{\mathrm{PINNs}}$ , testing loss $\mathrm{MSE}_{\infty}$ , and validation boundary loss $\mathrm{MSE}_{\partial\Omega}^{\mathrm{val}}$ , evaluated on the validation subset of boundary points. The dashed vertical line marks the checkpoint selected by the minimum validation boundary loss. (b) PINN prediction. (c) Exact solution.

Panel of Figure 5 — Figure 5: Aronsson example 6.2.2 on square domain. (a) Loss histories versus epoch: training boundary loss $\mathrm{MSE}_{\partial\Omega}$ , training PINNs loss $\mathrm{MSE}_{\mathrm{PINNs}}$ , testing loss $\mathrm{MSE}_{\infty}$ , and validation boundary loss $\mathrm{MSE}_{\partial\Omega}^{\mathrm{val}}$ , evaluated on the validation subset of boundary points. The dashed vertical line marks the checkpoint selected by the minimum validation boundary loss. (b) PINN prediction. (c) Exact solution.

	Simple training				Iterative training
$p$	total loss	$\text{MSE}_{\partial\Omega}$	PINNs loss	$\text{MSE}_{\infty}$	total loss	$\text{MSE}_{\partial\Omega}$	PINNs loss	$\text{MSE}_{\infty}$	$\text{MSE}_{\Delta_{\infty}}$	$\\|u_{\infty}\\|_{2,N_{\mathrm{test}}}$
$2$	$4.455$ e-05	$1.371$ e-05	$3.083$ e-03	$4.429$ e-03	$4.455$ e-05	$1.371$ e-05	$3.083$ e-03	$4.429$ e-03	$2.142$ e+01	$4.277$ e-01
$3$	$4.561$ e-05	$1.192$ e-05	$3.369$ e-03	$1.499$ e-03	$3.818$ e-05	$9.955$ e-06	$2.823$ e-03	$1.466$ e-03	$3.455$ e+00	$4.277$ e-01
$4$	$5.200$ e-07	$1.685$ e-07	$3.516$ e-03	$7.125$ e-04	$4.501$ e-05	$8.129$ e-06	$3.688$ e-03	$7.756$ e-04	$1.294$ e+00	$4.277$ e-01
$5$	$5.026$ e-07	$1.686$ e-07	$3.340$ e-03	$4.253$ e-04	$6.131$ e-05	$1.216$ e-05	$4.914$ e-03	$4.592$ e-04	$6.440$ e-01	$4.277$ e-01
$10$	$1.443$ e-05	$2.332$ e-06	$1.210$ e-03	$1.057$ e-04	$1.301$ e-04	$1.164$ e-05	$1.185$ e-02	$1.449$ e-04	$1.317$ e-01	$4.277$ e-01

Table 5: Comparison of simple and iterative PINNs training for the Aronsson example on the square domain. For both methods, we report the total training loss, boundary loss

\text{MSE}_{\partial\Omega}

, PINNs physical loss, and solution error

\text{MSE}_{\infty}

. For iterative training, we additionally report the residual error

\text{MSE}_{\Delta_{\infty}}

under the homogeneous infinity Laplacian operator.

Panel of Figure 7 — Figure 7: Iterative PINN continuation results over increasing values of $p$ . In both panels, the blue curve represents the total training loss and the orange curve represents $\mathrm{MSE}_{\infty}$ . In panel (a), the green curve represents $\mathrm{MSE}_{\Delta_{\infty}}$ for the Aronsson example. In panel (b), the green curve represents the direct finite- $p$ error $\mathrm{MSE}_{p}$ against the exact solution $u_{p}$ .

$p$	total loss	$\mathrm{MSE}_{\partial\Omega}$	PINNs loss	$\mathrm{MSE}_{\infty}$	$\\|u_{\infty}\\|_{2,N_{\mathrm{test}}}$	$\mathrm{MSE}_{p}$	$\\|u_{p}\\|_{2,N_{\mathrm{test}}}$
$2$	$1.287$ e-05	$1.203$ e-05	$8.403$ e-04	$7.145$ e-02	$4.090$ e-01	$8.861$ e-07	$1.446$ e-01
$3$	$1.930$ e-05	$9.691$ e-06	$9.607$ e-04	$3.070$ e-02	$4.090$ e-01	$1.645$ e-05	$2.395$ e-01
$4$	$1.366$ e-05	$7.840$ e-06	$5.816$ e-04	$1.696$ e-02	$4.090$ e-01	$2.982$ e-05	$2.851$ e-01
$5$	$1.173$ e-05	$4.237$ e-06	$7.498$ e-04	$1.033$ e-02	$4.090$ e-01	$1.284$ e-05	$3.116$ e-01
$10$	$1.090$ e-05	$1.296$ e-06	$9.608$ e-04	$2.507$ e-03	$4.090$ e-01	$9.213$ e-06	$3.620$ e-01
$50$	$5.152$ e-06	$2.383$ e-07	$4.914$ e-04	$9.981$ e-05	$4.090$ e-01	$1.740$ e-06	$3.999$ e-01

Table 6: Direct finite-

p

validation of the iterative PINN method for the distance-to-boundary problem on the unit disc. We report the total training loss, the boundary loss

\mathrm{MSE}_{\partial\Omega}

, the PINNs physical loss, the solution error

\mathrm{MSE}_{\infty}

against the limiting distance function

u_{\infty}

, and the direct solution error

\mathrm{MSE}_{p}

against the exact finite-

p

solution

u_{p}

		Disc domain			Square domain
$p$	$\text{MSE}_{\infty,p}^{\text{FEM}}$	$\text{MSE}_{\infty,p}$ (no limit reference)	$\text{MSE}_{\infty,p}$	$\text{MSE}_{\infty,p}^{\text{FEM}}$	$\text{MSE}_{\infty,p}$ (no limit reference)	$\text{MSE}_{\infty,p}$
$5$	$1.575$ e-02	$1.289$ e-02	$1.288$ e-02	$2.377$ e-02	$2.337$ e-02	$2.328$ e-02
$50$	$4.022$ e-05	$4.009$ e-05	$4.172$ e-05	$6.865$ e-05	$6.628$ e-05	$6.834$ e-05
$100$	$5.610$ e-06	$5.250$ e-06	$6.804$ e-06	$1.180$ e-05	$1.177$ e-05	$1.193$ e-05
$200$	$1.136$ e-06	$3.251$ e-06	$5.656$ e-06	$1.523$ e-06	$2.126$ e-06	$2.216$ e-06
$300$	$-$	$8.135$ e-05	$5.098$ e-05	$-$	$2.603$ e-05	$6.411$ e-06

Table 7: Error evaluation for the distance to origin DeepONet example 6.4.1 on the disc and square domains. For each

p

value, we show the FEM convergence-to-limit baseline

\text{MSE}_{\infty,p}^{\text{FEM}}

. We also show the DeepONet error

\text{MSE}_{\infty,p}

which compares against the exact

p\to\infty

limiting solution both without including

p=\infty

training data and with PINNs-predicted

p=\infty

data added as a surrogate for

p=500

Panel of Figure 8 — Figure 8: Distance to origin DeepONet example 6.4.1 (a) on unit disc domain (b) on square domain. The blue curves show the FEM convergence-to-limit baseline $\text{MSE}_{\infty,p}^{\text{FEM}}$ , the orange curves show the DeepONet error $\text{MSE}_{\infty,p}$ against the exact $p\to\infty$ limiting solution without including $p=\infty$ training data, and the green curves show the DeepONet error $\text{MSE}_{\infty,p}$ with the PINNs-predicted solution at $p=\infty$ added as a surrogate for $p=500$ . All DeepONet errors in this figure are measured against the limiting solution $u_{\infty}$ , not against the finite- $p$ FEM solution $u_{p}$ .

Panel of Figure 9 — Figure 9: Distance to boundary DeepONet results for various 2D domains. (a)–(d) show $\text{MSE}_{\infty,p}^{\text{FEM}}$ and $\text{MSE}_{\infty,p}$ over increasing $p$ values for: (a) the unit disc, (b) an elongated ellipse, (c) a rotated ellipse, and (d) a compressed ellipse. All predictions are compared against the $p=\infty$ solution which is the true distance function.

Panel of Figure 10 — Figure 10: Distance to boundary DeepONet results for various 3D domains. (a)–(c) show $\text{MSE}_{\infty,p}^{\text{FEM}}$ and $\text{MSE}_{\infty,p}$ over increasing $p$ values for: (a) the unit ball, (b) cylinder with radius $1$ and length $2$ , and (c) a torus with major radius $2$ and minor radius $1$ . All predictions are compared against the $p=\infty$ solution which is the true distance function.

Example	Equation of domain	$\\|u_{\infty}\\|_{2,N_{\mathrm{test}}}$	At largest training data $p$		Runtime
			$\text{MSE}_{\infty,p}^{\text{FEM}}$	$\text{MSE}_{\infty,p}$	DeepONet	Newton
			$\text{MSE}_{\infty,p}^{\text{FEM}}$	$\text{MSE}_{\infty,p}$	inference	Newton
2D disc, Fig 9 (a)	$x^{2}+y^{2}\leq 1$	$4.085$ e-01	$8.468$ e-06	$5.248$ e-06	$2.265$ e-04s	$8667$ s
Ellipse 1, Fig 9 (b)	$x^{2}+16y^{2}\leq 1$	$1.239$ e-01	$7.079$ e-06	$1.313$ e-05	$1.414$ e-04s	$\sim$ 3.5 hours
Ellipse 2, Fig 9 (c)	$8.5x^{2}+8.5y^{2}-15xy\leq 1$	$1.237$ e-01	$7.079$ e-06	$1.066$ e-05	$1.406$ e-04s
Ellipse 3, Fig 9 (d)	$4x^{2}+y^{2}\leq 1$	$2.398$ e-01	$5.711$ e-06	$4.435$ e-06	$1.416$ e-04s

Table 8: Distance to boundary DeepONet example 6.4.2 results across various 2D and 3D domains. The second column includes the equation for each domain and the third column shows normalized discrete norm of

u_{\infty}

. The fourth and fifth columns show

\text{MSE}_{\infty,p}^{\text{FEM}}

and

\text{MSE}_{\infty,p}

errors respectively, at the largest training

p

value. Lastly, the sixth and seventh columns show DeepONet inference times and total Newton FEM solver times. Both errors are measured against the limiting distance function, not against finite-

p

solutions.

$p$	Evaluation	$\mathrm{MSE}_{p}$	$\mathrm{MSE}_{\infty,p}$
$5$	Training	$5.340$ e-08	$9.704$ e-03
$7$	Interpolation	$1.584$ e-06	$4.940$ e-03
$10$	Training	$4.621$ e-08	$2.269$ e-03
$12$	Interpolation	$9.675$ e-08	$1.540$ e-03
$25$	Training	$6.696$ e-08	$3.492$ e-04
$37$	Interpolation	$7.756$ e-08	$1.562$ e-04

Table 9: Finite-

p

validation of the unit-disc DeepONet model. The error

\mathrm{MSE}_{p}

compares the DeepONet prediction with the exact finite-

p

solution

u_{p}

, while

\mathrm{MSE}_{\infty,p}

compares it with the limiting solution

u_{\infty}

Panel of Figure 11 — Figure 11: Distance to boundary DeepONet for all 2D ellipse examples. (a) $\overline{\mathrm{MSE}}(\theta)$ ( 34 ) as a function of the rotation angle $\theta$ , evaluated at $p=500$ . Blue points indicate $\theta$ values included in the training data. (b) $\overline{\mathrm{MSE}}(a,b)$ ( 35 ) over varying semi-axis lengths $a$ and $b$ , evaluated at $p=500$ . The DeepONet model is trained over sampled values of $(a,b,\theta)$ and tested on unseen configurations within the parameter range.

Panel of Figure 12 — Figure 12: Distance to boundary DeepONet for all 2D ellipse examples. (a)–(d) show $\overline{\mathrm{MSE}}(\theta)$ ( 34 ) using $3$ , $5$ , $7$ , and $9$ training $\theta$ values, respectively. Blue points indicate $\theta$ values included in the training data.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Table 21

Stabilization ablations
	Single-network ( $4\times 128$ reference)
Variant	Arctan	Aronsson (square)	Aronsson (disc)
normal (baseline)	$2.42$ e-07	$5.74$ e-07	$4.77$ e-07
clip0	$1.26$ e-07	$5.74$ e-07	$4.77$ e-07
out0	$1.84$ e-06	$6.17$ e-06	$1.00$ e-04
eta0	$9.47$ e-06	$4.01$ e-07	$3.14$ e-07

Appendix

Table 12: Ablation study: best-model test MSE for the infinity Laplacian PINNs.

Single component disabled
Configuration	$p=10$	$p=200$
baseline (all enabled)	$1.057$ e-04	$1.652$ e-02
clip0 (no residual clipping)	$1.057$ e-04	$3.159$ e-03
out0 (no outlier removal)	$1.539$ e-04	$1.692$ e-02
eta0 (no $\eta$ -regularization)	$1.057$ e-04	$1.652$ e-02
gradclip0 (no gradient norm clipping)	$1.931$ e-04	$-$

Appendix

Table 13: Ablation study for the simple

p

-Laplacian PINNs (Aronsson example on square domain). Each row disables one or more stabilization techniques. ”

-

” indicates that training diverged and therefore produced no finite MSE value.

Band	$p$ values	Epochs/ $p$	LR schedule	$\alpha$ schedule
1	$2,3,\dots,10,15,20$ (11 values)	100	Cosine $10^{-3}\!\to\!10^{-5}$	AlphaPlateauScheduler (adaptive)
2	$25,30,40,50$	20	Constant $10^{-5}$	Fixed $\alpha=10^{-2}$
3	$60,70,80,90,100$	20	Constant $10^{-5}$	Fixed $\alpha=10^{-3}$
4	$150,200,300,400,500$	20	Constant $10^{-5}$	Fixed $\alpha=10^{-4}$
5	$600,700,800,900,1000$	20	Constant $10^{-5}$	Fixed $\alpha=10^{-5}$

Appendix

Table 14: Five-band iterative training schedule for the homogeneous

p

-Laplacian experiments. Band 1 uses early stopping; bands 2–5 run for a fixed number of epochs per

p

value.

Stage	$p$ values	Epochs/ $p$	Interior points	LR schedule	$\alpha$ schedule
1	$2,3,\ldots,10$	up to $100$	$783{,}764$	Cosine $10^{-3}\!\to\!10^{-5}$	AlphaPlateauScheduler
2	$11,12,\ldots,20$	$50$	$195{,}496$	Cosine $10^{-4}\!\to\!10^{-5}$	Fixed $10^{-2}$
3	$20$ (refinement)	$50$	$783{,}764$	Cosine $10^{-4}\!\to\!10^{-5}$	Fixed $10^{-2}$
4	$25,30,40,50$	$50$	$195{,}496$	Cosine $10^{-4}\!\to\!10^{-5}$	Fixed $10^{-2}$
5	$60,70,80,90,100$	$50$	$195{,}496$	Cosine $10^{-4}\!\to\!10^{-5}$	Fixed $10^{-2}$
6	$150,200,300,400,500$	$50$	$195{,}496$	Constant $10^{-5}$	Fixed $10^{-2}$

Appendix

Table 15: Continuation schedule used for the direct finite-

p

iterative PINN validation on the unit disc. The experiment uses

10{,}000

boundary points. Stage 1 uses

8{,}000

training and

2{,}000

validation boundary points; subsequent stages use all

10{,}000

boundary points.

Domain	Interior	Boundary	Test	$w_{\text{pos}}$	Time (s)	Test MSE
Distance-to-origin (2D)
Disc	783,764	1,100	7,668	1.0	524	$5.948\times 10^{-7}$
Square	1,000,000	1,100	10,000	1.0	691	$2.985\times 10^{-7}$
Distance-to-boundary (2D)
Disc	783,764	10,000	7,668	1.0	523	$5.749\times 10^{-7}$
Square	1,000,000	40,000	10,000	1.0	681	$1.713\times 10^{-6}$

Appendix

Table 16: Eikonal PINNs training configuration per domain. Interior and boundary columns give the total number of points before the

80/20

train/validation split. Test MSE is the best-checkpoint mean squared error against the exact distance function on the listed test grid.

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The traditional limitations of neural networks in reliably generalizing beyond the convex hulls of their training data present a significant problem for computational physics, in which one often wishes to solve PDEs in regimes far beyond anything which can be experimentally or analytically validated. In this paper, we show how it is possible to circumvent these limitations by constructing formally-verified neural network solvers for PDEs, with rigorous convergence, stability, and conservation properties, whose correctness can therefore be guaranteed even in extrapolatory regimes. By using the method of characteristics to predict the analytical properties of PDE solutions a priori (even in regions arbitrarily far from the training domain), we show how it is possible to construct rigorous extrapolatory bounds on the worst-case L^inf errors of shallow neural network approximations. Then, by decomposing PDE solutions into compositions of simpler functions, we show how it is possible to compose these shallow neural networks together to form deep architectures, based on ideas from compositional deep learning, in which the large L^inf errors in the approximations have been suppressed. The resulting framework, called BEACONS (Bounded-Error, Algebraically-COmposable Neural Solvers), comprises both an automatic code-generator for the neural solvers themselves, as well as a bespoke automated theorem-proving system for producing machine-checkable certificates of correctness. We apply the framework to a variety of linear and non-linear PDEs, including the linear advection and inviscid Burgers' equations, as well as the full compressible Euler equations, in both 1D and 2D, and illustrate how BEACONS architectures are able to extrapolate solutions far beyond the training data in a reliable and bounded way. Various advantages of the approach over the classical PINN approach are discussed.

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