Internal variable reformulation of Volterra integro-differential equations with exponential kernels for physics-informed neural networks

Yongseok Jang; Yongseok Jang

doi:10.3934/math.2026677

AIMS Mathematics

2026, Volume 11, Issue 6: 16511-16533. doi: 10.3934/math.2026677

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Internal variable reformulation of Volterra integro-differential equations with exponential kernels for physics-informed neural networks

Yongseok Jang ^,

Department of Mathematics, Chonnam National University, Gwangju, 61186, South Korea

Received: 30 March 2026 Revised: 04 May 2026 Accepted: 19 May 2026 Published: 09 June 2026
MSC : 45D05, 65N75, 68T20, 74D05

We developed a physics-informed neural network (PINN) framework for solving integro-differential equations (IDEs), with particular emphasis on Volterra-type problems with exponentially decaying kernels. While PINNs provide a flexible approach for incorporating physical laws without mesh-based discretization, the treatment of convolution integral terms remains computationally demanding. To address this issue, we introduced internal variables for exponential kernels, transforming the original IDE into an equivalent system of differential equations, eliminating the need for explicit quadrature, and significantly reducing computational cost and memory requirements. The proposed method incorporates both differential and integral operators within the PINN framework. Numerical results demonstrate that the method maintains accuracy while significantly improving computational efficiency. It also extends naturally to inverse problems, where viscoelastic parameters are accurately identified from sparse observations.
- physics-informed neural network,
- integro-differential equation,
- exponential kernels,
- internal variables,
- viscoelasticity
Citation: Yongseok Jang. Internal variable reformulation of Volterra integro-differential equations with exponential kernels for physics-informed neural networks[J]. AIMS Mathematics, 2026, 11(6): 16511-16533. doi: 10.3934/math.2026677

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Abstract

We developed a physics-informed neural network (PINN) framework for solving integro-differential equations (IDEs), with particular emphasis on Volterra-type problems with exponentially decaying kernels. While PINNs provide a flexible approach for incorporating physical laws without mesh-based discretization, the treatment of convolution integral terms remains computationally demanding. To address this issue, we introduced internal variables for exponential kernels, transforming the original IDE into an equivalent system of differential equations, eliminating the need for explicit quadrature, and significantly reducing computational cost and memory requirements. The proposed method incorporates both differential and integral operators within the PINN framework. Numerical results demonstrate that the method maintains accuracy while significantly improving computational efficiency. It also extends naturally to inverse problems, where viscoelastic parameters are accurately identified from sparse observations.

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