Analysis of a $(2 + 1)$-dimensional fractal KP model: solitons, geometric embedding, and parameter estimation via deep learning

This paper presents fractal-modified $(2 + 1)$-dimensional Kadomtsev–Petviashvili (KP) model for describing long waves in fractal dispersive media, based on He’s ($\mathfrak{H}$) fractal derivative. To begin with, we implement the semi-inverse technique (SIT) to formulate a fractal generalized variational principle (FGVP), and we prove it using a fractal two-scale transformation and a newly derived principle. The conservation laws in the fractal space are expressed in energy form. The Bäcklund transformation-based approach proposed by Wang, combined with symbolic computation and ansatz function schemes, enables the derivation of further solutions. These solutions include new forms of rational functions, double exponential functions, sine-cosine forms, and hyperbolic sine-cosine functions. For all these solutions, the effects of fractal orders are thoroughly analyzed using 3D plots, density plots, and 2D curves. Additionally, a feedforward deep neural network is trained to estimate the fractal parameters $\zeta _{1}$ and γ from waveform data. Furthermore, manifold learning techniques such as principal component analysis and uniform manifold approximation and projection are employed to uncover global structural patterns in the solution space, revealing distinct clusters and nonlinear dependencies driven by fractal parameters.