https://doi.org/10.1140/epjp/s13360-025-06952-3
Regular Article
Single-layer bilinear neural network modeling of lump and multi-soliton interactions for the Kadomtsev–Petviashvili system
Department of Mathematics, COMSATS University Islamabad, Lahore, Lahore Campus, Pakistan
a
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Received:
30
August
2025
Accepted:
9
October
2025
Published online:
25
October
2025
Abstract
This paper presents a bilinear neural network modeling (BNNM) framework that couples the Hirota bilinear approach with trainable neural parameters to obtain closed-form solutions of the (3+1)-dimensional generalized Kadomtsev–Petviashvili equation (gKPE). Throughout, we employ a [4-3-1] neural network model—three layers comprising a 4-neuron input layer, a 3-neuron hidden layer, and a 1-neuron output. The method automatically identifies soliton parameters and yields exact expressions for lump waves, breathers, and one-, two-, and multi-soliton interactions. These solutions capture localized pulses, oscillatory packets, and collision dynamics relevant to shallow water and internal waves, plasmas, and atmospheric flows. Compared with using Hirota alone, BNNM preserves analytic interpretability while improving parameter discovery and verification through residual error diagnostics. Visual and quantitative evaluations (surface/contour plots and residual statistics) confirm the correctness of the solutions. Overall, BNNM provides a compact, reproducible workflow for high-dimensional nonlinear PDEs that balance symbolic structure with data-driven optimization.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2025
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
