Simultaneous imposition of initial and boundary conditions via decoupled physics-informed neural networks for solving initial-boundary value problems

doi:10.1007/s10483-025-3240-7

Abstract

Abstract:

Enforcing initial and boundary conditions (I/BCs) poses challenges in physics-informed neural networks (PINNs). Several PINN studies have gained significant achievements in developing techniques for imposing BCs in static problems; however, the simultaneous enforcement of I/BCs in dynamic problems remains challenging. To overcome this limitation, a novel approach called decoupled physics-informed neural network (dPINN) is proposed in this work. The dPINN operates based on the core idea of converting a partial differential equation (PDE) to a system of ordinary differential equations (ODEs) via the space-time decoupled formulation. To this end, the latent solution is expressed in the form of a linear combination of approximation functions and coefficients, where approximation functions are admissible and coefficients are unknowns of time that must be solved. Subsequently, the system of ODEs is obtained by implementing the weighted-residual form of the original PDE over the spatial domain. A multi-network structure is used to parameterize the set of coefficient functions, and the loss function of dPINN is established based on minimizing the residuals of the gained ODEs. In this scheme, the decoupled formulation leads to the independent handling of I/BCs. Accordingly, the BCs are automatically satisfied based on suitable selections of admissible functions. Meanwhile, the original ICs are replaced by the Galerkin form of the ICs concerning unknown coefficients, and the neural network (NN) outputs are modified to satisfy the gained ICs. Several benchmark problems involving different types of PDEs and I/BCs are used to demonstrate the superior performance of dPINN compared with regular PINN in terms of solution accuracy and computational cost.

Key words: decoupled physics-informed neural network (dPINN), decoupled formulation, Galerkin method, initial-boundary value problem (IBVP), machine learning

2010 MSC Number:

O241

K. A. LUONG, M. A. WAHAB, J. H. LEE. Simultaneous imposition of initial and boundary conditions via decoupled physics-informed neural networks for solving initial-boundary value problems. Applied Mathematics and Mechanics (English Edition), 2025, 46(4): 763-780.

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Figures/Tables 13

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Table 1

Comparison of training time (in minutes) of the dPINN and PINN"

Problem	$t$ /min
Problem	dPINN	PINN
Heat transfer (see Subsection 3.1)	0.62	62.43
Wave propagation (see Subsection 3.2)	1.11	101.55
Cantilever bar (see Subsection 3.3)	1.43	53.37
Burgers' equation (see Subsection 3.4)	0.84	46.11
Advection-diffusion equation (see Subsection 3.5)	0.80	192.69
Fixed-beam (see Subsection 3.6)	0.70	180.92

Table 1

Fig. 11

Fig. A1

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