Randomized Neural Networks for Partial Differential Equation on Static and Evolving Surfaces

Jingbo Sun; Fei Wang

math.NAMar 2, 2026

Randomized Neural Networks for Partial Differential Equation on Static and Evolving Surfaces

Authors: Jingbo Sun, Fei Wang

Organizations: School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, P.R. China. · School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China.

Abstract

Surface partial differential equations arise in numerous scientific and engineering applications. Their numerical solution on static and evolving surfaces remains challenging due to geometric complexity and, for evolving geometries, the need for repeated mesh updates and geometry or solution transfer. While neural-network-based methods offer mesh-free discretizations, approaches based on nonconvex training can be costly and may fail to deliver high accuracy in practice. In this work, we develop a randomized neural network (RaNN) method for solving PDEs on both static and evolving surfaces: the hidden-layer parameters are randomly generated and kept fixed, and the output-layer coefficients are determined efficiently by solving a least-squares problem. For static surfaces, we present formulations for parametrized surfaces, implicit level-set surfaces, and point-cloud geometries, and provide a corresponding theoretical analysis for the parametrization-based formulation with interface compatibility. For evolving surfaces with topology preserved over time, we introduce a RaNN-based strategy that learns the surface evolution through a flow-map representation and then solves the surface PDE on a space--time collocation set, avoiding remeshing. Extensive numerical experiments demonstrate broad applicability and favorable accuracy--efficiency performance on representative benchmarks.

Figures & tables

Figure 1: Architecture of Randomized Neural Networks

$M$	600		1000		1400
$N_{\rm col}$	Error	Time (s)	Error	Time (s)	Error	Time (s)
900	1.77E-04	0.078	2.95E-04	0.139	4.17E-04	0.191
2500	4.77E-05	0.111	4.04E-06	0.213	1.63E-06	0.304
4900	1.79E-05	0.172	2.06E-07	0.449	1.32E-07	0.721
8100	3.86E-05	0.415	3.66E-07	0.666	5.90E-08	1.018

Table 1: Relative errors obtained by the single-chart parametrization-based approach (with periodic boundary compatibility enforced) in Example 5.1

$M$	600		1000		1400
$N_{\rm col}$	Error	Time (s)	Error	Time (s)	Error	Time (s)
900	4.23E-07	0.081	1.33E-07	0.143	1.01E-07	0.178
2500	4.16E-11	0.118	4.74E-12	0.229	3.34E-12	0.400
4900	1.01E-10	0.262	1.93E-12	0.548	4.87E-14	1.015
8100	7.61E-11	0.597	7.11E-13	1.072	2.57E-13	1.626

Table 2: Relative errors obtained by parametrization-based approach (no separate boundary enforcement) in Example 5.1 .

$M$	1200		1600		2000
$N_{\rm col}$	Error	Time (s)	Error	Time (s)	Error	Time (s)
2000	5.70E-07	0.229	1.18E-06	0.366	7.98E-08	0.485
4000	1.16E-08	0.334	1.25E-06	0.740	6.42E-08	0.984
6000	5.57E-07	0.567	3.01E-07	0.988	3.51E-08	1.387
8000	6.43E-08	0.770	9.04E-08	1.344	1.29E-08	1.755

Table 3: Relative errors (evaluated on

P_{CL}^{\rm coarse}

) obtained by the level-set-based approach in Example 5.2 .

$M$	1200		1600		2000
$N_{\rm col}$	Error	Time (s)	Error	Time (s)	Error	Time (s)
2000	1.44E-09	0.236	6.93E-10	0.331	1.36E-10	0.456
4000	1.04E-09	0.427	1.08E-10	0.606	3.72E-12	0.991
6000	3.30E-10	0.690	1.83E-11	1.013	3.60E-12	1.285
8000	3.26E-10	0.900	1.19E-11	1.308	1.76E-12	1.702

Table 4: Relative errors (evaluated on

P_{CL}^{\rm fine}

) obtained by the level-set-based approach in Example 5.2 .

Figure 2: Pointwise absolute errors of the two point sets.

$M$	1200		1600		2000		2400
$N_{\rm col},\ N_{0}$	Error	Time (s)	Error	Time (s)	Error	Time (s)	Error	Time (s)
2000, 1000	1.89E-04	0.280	1.40E-04	0.428	1.56E-04	0.655	1.45E-04	1.137
4000, 1000	9.97E-05	0.642	4.94E-05	0.901	2.11E-05	1.204	1.21E-05	1.598
6000, 1000	4.55E-05	0.915	1.89E-05	1.285	1.36E-05	1.571	5.90E-06	2.098
8000, 1000	4.80E-05	1.095	1.52E-05	1.587	9.78E-06	1.985	3.43E-06	2.541

Table 5: Relative errors obtained by the level-set-based approach using

\mathring{P}^{\rm coarse}_{CL}

in Example 5.3 .

$M$	1200		1600		2000		2400
$N_{\rm col},\ N_{0}$	Error	Time (s)	Error	Time (s)	Error	Time (s)	Error	Time (s)
2000, 1000	1.89E-04	0.271	1.37E-04	0.477	1.65E-04	0.667	1.50E-04	0.938
4000, 1000	9.95E-05	0.534	4.94E-05	0.834	1.97E-05	1.281	1.13E-05	1.637
6000, 1000	4.55E-05	0.927	1.91E-05	1.231	1.29E-05	1.672	5.23E-06	2.111
8000, 1000	4.80E-05	1.114	1.52E-05	1.582	9.71E-06	2.050	3.57E-06	2.540

Table 6: Relative errors obtained by the level-set-based approach using

\mathring{P}^{\rm fine}_{CL}

in Example 5.3 .

Figure 3: Predicted temperature distribution on the cup-shaped surface.

Figure 4: Predicted heat distribution on the bunny surface.

$M_{1},M_{2},M_{3}$	$N_{\rm col},N_{x}$	$E_{\mbox{\boldmath{$ x $}}}$	$E_{\mbox{\boldmath{$ n $}}}$	$E_{H}$	Time (s)
1600,500,500	$80^{2},80$	2.27E-06	1.58E-06	2.42E-06	0.624
	$120^{2},120$	7.48E-08	1.17E-07	6.62E-07	1.324
	$160^{2},160$	8.06E-09	8.63E-08	6.47E-07	2.150
	$200^{2},200$	5.31E-09	3.64E-08	1.81E-07	3.510
2600,500,500	$80^{2},80$	5.87E-05	1.11E-05	1.71E-05	1.125
	$120^{2},120$	1.32E-07	1.27E-07	1.88E-07	2.136

Table 7: Relative errors of

x

n

, and

H

t=2

and training time for the learned surface evolution.

Figure 5: Evolving ellipsoid and corresponding mean curvature distribution.

$M$	1400		2000		2600
$N_{x},N_{\rm col}$	Error	Time (s)	Error	Time (s)	Error	Time (s)
$80,80^{2}$	2.41E-04	1.140	9.63E-05	1.714	7.80E-05	2.370
$120,120^{2}$	2.83E-05	2.218	8.16E-06	3.301	1.03E-05	4.484
$160,160^{2}$	4.66E-06	3.860	3.05E-06	5.613	1.40E-06	7.532
$200,200^{2}$	3.32E-06	5.901	6.97E-07	8.579	3.07E-07	11.330

Table 8: Relative errors of

u

t=2

on the evolving surface and training time.

$M_{1},M_{2},M_{3}$	$N_{\rm col},N_{0}$	$E_{\mbox{\boldmath{$ x $}}}$	$E_{\mbox{\boldmath{$ n $}}}$	$E_{H}$	Time (s)
1600	$80^{2},80$	1.11E-06	7.49E-06	3.36E-05	0.975
	$120^{2},120$	8.43E-10	7.98E-09	5.50E-08	2.046
	$160^{2},160$	1.19E-09	1.19E-08	6.50E-08	3.754
	$200^{2},200$	8.44E-10	1.19E-08	8.77E-08	6.137
2600	$80^{2},80$	1.90E-06	1.43E-05	6.02E-05	1.805
	$120^{2},120$	2.18E-08	2.22E-07	1.57E-06	3.685

Table 9: Relative errors of

x

n

, and

H

t=3

and training time.

Figure 6: Time histories of droplet volume error (a) and surfactant mass error (b).

Figure 7: Snapshots of the droplet surface

\mathcal{N}^{1,2,3}_{e}

and surfactant concentration

u_{e}

at different times. Colors indicate the magnitude of

u_{e}

Neural PDE Solvers

Neural PDE solvers use trainable models to approximate solutions or solution operators for partial differential equations, often with physical constraints.

See all 269 papers

Adaptive-Distribution Randomized Neural Networks for PDEs: A Low-Dimensional Distribution-Learning Framework
Apr 27, 2026You Yang, Fei WangNeural PDE Solvers PDE Operator Learning
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China. · School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China.
Random Neural Network Expressivity for Non-Linear Partial Differential Equations
May 24, 2026Muhammed Ali Mehmood, Lukas GononNeural PDE Solvers PDE Solving
Department of Mathematics Imperial College London UK · School of Computer Science University of St. Gallen Switzerland
Adaptive Randomized Neural Networks for PDEs: Structure-Aware Feature Redistribution and Residual-Driven Block Enrichment
Oct 15, 2026You Yang, Fei WangNeural PDE Solvers Random Feature Methods
School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China. · School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China.

Apr 27, 2026J/K move · Enter open · S save

Adaptive-Distribution Randomized Neural Networks for PDEs: A Low-Dimensional Distribution-Learning Framework

You Yang, Fei Wang

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China. · School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China.

Neural PDE Solvers PDE Operator Learning

Open paper AskPDF arXiv

Randomized neural networks (RaNNs) are attractive for partial differential equations (PDEs) because they replace expensive end-to-end training with a linear least-squares solve over randomized hidden features. Their practical performance, however, depends strongly on the sampling distribution of the hidden-layer parameters, which is usually chosen heuristically and problem by problem. This distribution sensitivity is a central bottleneck in randomized neural PDE solvers. In this work, we propose Adaptive-Distribution Randomized Neural Networks (AD-RaNN), a framework that promotes randomized feature generation from a fixed heuristic choice to a low-dimensional adaptive optimization problem. Instead of training all hidden weights and biases, AD-RaNN parameterizes the hidden-feature sampling distribution by a low-dimensional vector p and optimizes only p, thereby preserving the least-squares structure of RaNNs while reducing manual distribution tuning. The method uses a two-stage strategy: ridge-regularized reduced training for stable distribution-parameter optimization, followed by an unregularized least-squares refit for final solution recovery. We develop two adaptive mechanisms, PDE-Driven Adaptive Distribution (PDAD) and Data-Driven Adaptive Distribution (DDAD), and deploy them in space-time solvers, discrete-time solvers, and operator-learning models. We also incorporate an adaptive layer-growth enhancement for localized structures. For the reduced optimization problem, we establish well-posedness of the reduced objectives, consistency of ridge-regularized minimizers, an efficient gradient formula, and a practical lower-bound estimate for the ridge parameter. Numerical experiments on benchmark problems show that AD-RaNN provides an effective distribution-level adaptation mechanism, reduces reliance on hand-crafted hidden-feature distributions, and achieves strong empirical accuracy.

Abstract

Neural PDE Solvers

Explore similar work

Adaptive-Distribution Randomized Neural Networks for PDEs: A Low-Dimensional Distribution-Learning Framework

Random Neural Network Expressivity for Non-Linear Partial Differential Equations

Adaptive Randomized Neural Networks for PDEs: Structure-Aware Feature Redistribution and Residual-Driven Block Enrichment