Physics-informed neural networks with exponential basis augmentation for nonlinear wave solutions of the (2+1)-dimensional Zakharov Kuznetsov equation

A rich array of dynamical structures, including localized solitons, periodic wave trains, and breather-type oscillations, arises from the nonlinear partial differential equations governing the propagation of waves in multiple spatial dimensions. The current work presents and comparatively evaluates two neural surrogate frameworks, labeled PINN and E-PINN for continuity with their solution-informed physical grounding even though neither enforces the governing equation as an explicit residual term. These frameworks are designed to approximate solutions of the ( 2 + 1 ) -dimensional Zakharov–Kuznetsov (ZK) equation: a baseline neural approximator trained directly on analytical target data, and an exponential-augmented counterpart (E-PINN) in which a structured exponential basis function, derived from the traveling-wave geometry of the ZK equation, is embedded as a fixed auxiliary predictor alongside the trainable network. To study the effects of solution complexity on approximation quality, three representative, physically motivated, exponentially structured benchmark waveforms – soliton, periodic, and breather-type – are adopted as benchmark targets, allowing a systematic assessment of accuracy, training convergence, and model reliability across a range of nonlinearity and oscillatory behavior. The computed results reveal a distinctly solution-specific performance profile. For localized soliton solutions, whose spatial structure is naturally exponential, the augmented framework attains a peak-region error reduction of up to 24% relative to the baseline, demonstrating that incorporating a structurally compatible basis function simplifies the residual learning task. For periodic and breather-type solutions, however, the standard neural approximator proves more effective across all domain-averaged error measures – and the augmented variants’ peak error rises to between roughly 1.3 and 2.0 times the baseline value depending on the case – as the exponential basis introduces structural incompatibility rather than benefit in these regimes. Taken together, the findings indicate that architectural decisions driven by the mathematical form of the target solution yield selective gains rather than across-the-board improvements, and that principled model selection should be informed by prior knowledge of the dominant solution features. The proposed computational pipeline provides a clear and fully reproducible base for benchmarking neural approximation strategies in nonlinear dispersive wave systems.