Physics-informed neural network method for elasticity interface problems and its error analysis

Bokyu Kim; Dongsik Jo; Gwanghyun Jo; Bokyu Kim; Dongsik Jo; Gwanghyun Jo

doi:10.3934/math.20261218

AIMS Mathematics

2026, Volume 11, Issue 9: 30759-30781. doi: 10.3934/math.20261218

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Physics-informed neural network method for elasticity interface problems and its error analysis

Bokyu Kim ¹,
Dongsik Jo ^{2
,
,},
Gwanghyun Jo ^{1
,
,}

1.
Department of Mathematical Data Science, Hanyang University ERICA, Ansan-si 15588, Republic of Korea
2.
Department of Electrical, Electronic and Computer Engineering, University of Ulsan, Ulsan 44610, Republic of Korea

Received: 29 July 2026 Revised: 03 September 2026 Accepted: 08 September 2026 Published: 21 September 2026
MSC : 65N12, 65N15, 65N30

The elasticity problem describes the relationship between displacement and stress in an elastic body in response to an external force. However, when a spring-type jump condition is imposed at the interface, it is difficult to numerically approximate the solution because the displacement becomes discontinuous across the interface and the displacement jump is given only implicitly. We propose a physics-informed neural network (PINN) to approximate solutions of elasticity equations with spring-type jump conditions. Since a PINN is represented by a continuous neural network function, it is inefficient for directly approximating discontinuous solutions. To remedy this, we introduce an additional variable that labels the subdomains, thereby representing the discontinuous solution as a continuous function in an augmented higher-dimensional space. This continuous representation enables us to exploit the universal approximation property of neural networks. The loss function of the proposed PINN is constructed to incorporate the governing elasticity equations. We show that the energy-norm error of the neural network approximation can be controlled by the residuals of the loss function and prove that these residuals can be made arbitrarily small. In addition, we conduct numerical experiments for circular- and line-interface problems as well as a driven cavity problem. Comparisons with a piecewise PINN and an immersed finite element method demonstrate the accuracy and efficiency of the proposed method.
- elasticity interface problem,
- physics-informed neural network,
- nonhomogeneous interface,
- spring-type jump condition,
- error estimates
Citation: Bokyu Kim, Dongsik Jo, Gwanghyun Jo. Physics-informed neural network method for elasticity interface problems and its error analysis[J]. AIMS Mathematics, 2026, 11(9): 30759-30781. doi: 10.3934/math.20261218

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Abstract

The elasticity problem describes the relationship between displacement and stress in an elastic body in response to an external force. However, when a spring-type jump condition is imposed at the interface, it is difficult to numerically approximate the solution because the displacement becomes discontinuous across the interface and the displacement jump is given only implicitly. We propose a physics-informed neural network (PINN) to approximate solutions of elasticity equations with spring-type jump conditions. Since a PINN is represented by a continuous neural network function, it is inefficient for directly approximating discontinuous solutions. To remedy this, we introduce an additional variable that labels the subdomains, thereby representing the discontinuous solution as a continuous function in an augmented higher-dimensional space. This continuous representation enables us to exploit the universal approximation property of neural networks. The loss function of the proposed PINN is constructed to incorporate the governing elasticity equations. We show that the energy-norm error of the neural network approximation can be controlled by the residuals of the loss function and prove that these residuals can be made arbitrarily small. In addition, we conduct numerical experiments for circular- and line-interface problems as well as a driven cavity problem. Comparisons with a piecewise PINN and an immersed finite element method demonstrate the accuracy and efficiency of the proposed method.

References

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