Neural boundary shape functions in physics-informed neural networks for discontinuous and high-frequency problems

doi:10.1007/s10483-026-3350-8

Abstract

Abstract:

Physics-informed neural networks (PINNs) have been shown as powerful tools for solving partial differential equations (PDEs) by embedding physical laws into the network training. Despite their remarkable results, complicated problems such as irregular boundary conditions (BCs) and discontinuous or high-frequency behaviors remain persistent challenges for PINNs. For these reasons, we propose a novel two-phase framework, where a neural network is first trained to represent shape functions that can capture the irregularity of BCs in the first phase, and then these neural network-based shape functions are used to construct boundary shape functions (BSFs) that exactly satisfy both essential and natural BCs in PINNs in the second phase. This scheme is integrated into both the strong-form and energy PINN approaches, thereby improving the quality of solution prediction in the cases of irregular BCs. In addition, this study examines the benefits and limitations of these approaches in handling discontinuous and high-frequency problems. Overall, our method offers a unified and flexible solution framework that addresses key limitations of existing PINN methods with higher accuracy and stability for general PDE problems in solid mechanics.

Key words: physics-informed neural network (PINN), boundary shape function (BSF), strong-form approach, energy approach, discontinuity, high-frequency problem

2010 MSC Number:

P. T. NGUYEN, K. A. LUONG, J. H. LEE. Neural boundary shape functions in physics-informed neural networks for discontinuous and high-frequency problems. Applied Mathematics and Mechanics (English Edition), 2026, 47(2): 423-442.

TrendMD

Figures/Tables 27

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Table 1

Relative errors of approximated deflections obtained by the proposed methods with various activation functions and optimizers for the cantilever beam"

Activation function	Method	Optimizer
Activation function	Method	Adam	L-BFGS	Adagrad	Adam+L-BFGS
$Tanh$	Present-SF	$2.4 × 10 − 3$	$6.3 × 10 − 6$	$2.2 × 10 − 2$	$2.5 × 10 − 6$
$Tanh$	Present-En	$2.7 × 10 − 4$	$2.3 × 10 − 4$	$1.3 × 10 − 3$	$2.3 × 10 − 4$
$Sine$	Present-SF	$2.9 × 10 − 4$	$2.7 × 10 − 6$	$5.1 × 10 − 4$	$3.9 × 10 − 6$
$Sine$	Present-En	$2.8 × 10 − 4$	$2.4 × 10 − 4$	$2.8 × 10 − 4$	$2.4 × 10 − 4$
$Sigmoid$	Present-SF	$2.6 × 10 − 3$	$7.5 × 10 − 5$	$2.5 × 10 − 1$	$2.2 × 10 − 5$
$Sigmoid$	Present-En	$2.2 × 10 − 3$	$2.2 × 10 − 4$	$4.5 × 10 − 3$	$2.4 × 10 − 4$
$Relu$	Present-SF	$1.5 × 100$	$1.6 × 100$	$1.7 × 100$	$1.6 × 100$
$Relu$	Present-En	$> 102$	$> 102$	$> 102$	$> 102$
Note: Adam represents the adaptive moment estimation, L-BFGS represents the limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm, and Adagrad represents the adaptive gradient algorithm

Table 1

Table 2

Relative errors of approximated deflections obtained by the proposed methods with various network architectures for the cantilever beam"

Method	Two hidden layers			Three hidden layers
Method	30 neurons	40 neurons	50 neurons	30 neurons	40 neurons	50 neurons
Present-SF	$6.5 × 10 − 6$	$7.9 × 10 − 6$	$6.7 × 10 − 6$	$2.5 × 10 − 6$	$8.2 × 10 − 6$	$9.2 × 10 − 6$
Present-En	$2.3 × 10 − 4$	$1.9 × 10 − 4$	$2.3 × 10 − 4$	$2.3 × 10 − 4$	$2.1 × 10 − 4$	$2.1 × 10 − 4$

Table 2

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Problem	Present method			PINN-SF	PINN-En
Problem	Finding shape functions	Present-SF	Present-En	PINN-SF	PINN-En
Subsection 4.1.1	1.3	8.6	5.1	1.9	1.5
Subsection 4.1.2	16.6	44.7	8.2	9.3	2.2
Subsection 4.2.1	4.7	10.5	5.3	2.6	1.2
Subsection 4.2.2	20.3	–	17.2	–	3.7
Subsection 4.3.1	1.4	9.4	8.3	2.6	1.5
Subsection 4.3.2	16.6	71.2	24.1	17.9	5.4