Solving an inverse source problem for nonlocal diffusion-wave equations through Laplace-based physics-informed neural networks

Weiyue Sun; Changxin Qiu; Zhiyuan Li; Weiyue Sun; Changxin Qiu; Zhiyuan Li

doi:10.3934/era.2026056

Electronic Research Archive

2026, Volume 34, Issue 2: 1209-1237. doi: 10.3934/era.2026056

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Solving an inverse source problem for nonlocal diffusion-wave equations through Laplace-based physics-informed neural networks

Weiyue Sun ¹,
Changxin Qiu ^{2
,
,},
Zhiyuan Li ^{2
,
,}

1.
School of Accounting, Ningbo University of Finance and Economics, Ningbo 315175, China
2.
School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China

Received: 08 January 2026 Revised: 28 January 2026 Accepted: 03 February 2026 Published: 06 February 2026

This work investigates the nonlocal diffusion-wave equation governed by the time-fractional Caputo derivative. We first establish that the solution is $ t $-analytic through the application of Fourier expansion and the properties of the Mittag-Leffler functions. Building on this, we apply the Laplace transform to demonstrate the subordination principle for solutions of parabolic and hyperbolic equations in the context of the nonlocal diffusion-wave equation. Second, we prove the uniqueness and conditional stability of a solution to an inverse problem involving the determination of the spatially varying source term based on interior information from a subdomain. We alslo introduce a novel framework termed Laplace-based physics-informed neural networks (L-PINNs), which is tailored for determining source terms in nonlocal diffusion-wave systems. We substantiate the proposed approach through a series of numerical experiments, demonstrating its superior accuracy and computational efficiency.
- nonlocal diffusion-wave equation,
- inverse source problem,
- subordination principle,
- Laplace transform,
- Laplace-based physics-informed neural networks
Citation: Weiyue Sun, Changxin Qiu, Zhiyuan Li. Solving an inverse source problem for nonlocal diffusion-wave equations through Laplace-based physics-informed neural networks[J]. Electronic Research Archive, 2026, 34(2): 1209-1237. doi: 10.3934/era.2026056

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Abstract

This work investigates the nonlocal diffusion-wave equation governed by the time-fractional Caputo derivative. We first establish that the solution is $ t $-analytic through the application of Fourier expansion and the properties of the Mittag-Leffler functions. Building on this, we apply the Laplace transform to demonstrate the subordination principle for solutions of parabolic and hyperbolic equations in the context of the nonlocal diffusion-wave equation. Second, we prove the uniqueness and conditional stability of a solution to an inverse problem involving the determination of the spatially varying source term based on interior information from a subdomain. We alslo introduce a novel framework termed Laplace-based physics-informed neural networks (L-PINNs), which is tailored for determining source terms in nonlocal diffusion-wave systems. We substantiate the proposed approach through a series of numerical experiments, demonstrating its superior accuracy and computational efficiency.

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