<p>This paper presents a computational framework based on wavelet-enhanced Physics-Informed Neural Networks (PINNs) for solving nonlinear coupled ordinary differential equations arising in complex fluid transport problems. The framework is applied to the three-dimensional squeezing and rotating double-diffusive flow of a hybrid carbon nanotube (CNT) nanofluid in a parallel-plate channel incorporating cross-diffusion and internal heat source effects. The hybrid nanofluid consists of single-walled and multi-walled CNTs dispersed in water, while the mathematical model additionally accounts for chemical reaction, internal heat generation or absorption, and Soret-Dufour cross-diffusion mechanisms. Using similarity transformations, the governing nonlinear partial differential equations are reduced to a coupled system of nonlinear ordinary differential equations. A physics-informed neural network is then developed in which the governing equations and boundary conditions are embedded directly into the loss function, enabling a mesh-free solution procedure without labelled training data. To examine convergence and approximation behaviour, three wavelet-inspired activation functions Gaussian, Morlet, and Mexican Hat are incorporated into the network architecture and compared with conventional activations, namely <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\tanh \)</EquationSource> </InlineEquation> and Swish, under identical training conditions across ten independent runs. Among the examined configurations, the Gaussian activation produced the lowest mean residual loss together with comparatively stable convergence behaviour under the adopted training configuration. The computational framework is further assessed through residual analysis, sensitivity studies with respect to collocation density and network width, and comparison with numerical solutions obtained independently using the Haar Wavelet Collocation Method (HWCM). The two approaches show close numerical consistency across the similarity domain. In addition, a parametric study is conducted to investigate the effects of the squeezing parameter, rotation parameter, Soret and Dufour numbers, heat generation parameter, chemical reaction parameter, and hybrid CNT volume fraction on the velocity, temperature, and concentration fields, together with the associated skin-friction coefficient, Nusselt number, and Sherwood number. The results suggest that, under the adopted computational configuration, wavelet-inspired activation functions can provide comparatively more consistent convergence behaviour than conventional activations for the present strongly coupled nonlinear system. The proposed framework offers a mesh-free computational approach for boundary value problems governed by coupled nonlinear differential equations, while the reported observations are specific to the adopted training configuration and are not intended to imply universal performance across all PINN architectures or problem settings.</p>

A Wavelet-enhanced PINN framework for double-diffusive hybrid CNT nanofluid flow with cross-diffusion and heat generation

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发表日期：

2026年7月11日

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Govind Gaur,
D. Srinivasacharya

摘要

This paper presents a computational framework based on wavelet-enhanced Physics-Informed Neural Networks (PINNs) for solving nonlinear coupled ordinary differential equations arising in complex fluid transport problems. The framework is applied to the three-dimensional squeezing and rotating double-diffusive flow of a hybrid carbon nanotube (CNT) nanofluid in a parallel-plate channel incorporating cross-diffusion and internal heat source effects. The hybrid nanofluid consists of single-walled and multi-walled CNTs dispersed in water, while the mathematical model additionally accounts for chemical reaction, internal heat generation or absorption, and Soret-Dufour cross-diffusion mechanisms. Using similarity transformations, the governing nonlinear partial differential equations are reduced to a coupled system of nonlinear ordinary differential equations. A physics-informed neural network is then developed in which the governing equations and boundary conditions are embedded directly into the loss function, enabling a mesh-free solution procedure without labelled training data. To examine convergence and approximation behaviour, three wavelet-inspired activation functions Gaussian, Morlet, and Mexican Hat are incorporated into the network architecture and compared with conventional activations, namely \(\tanh \) and Swish, under identical training conditions across ten independent runs. Among the examined configurations, the Gaussian activation produced the lowest mean residual loss together with comparatively stable convergence behaviour under the adopted training configuration. The computational framework is further assessed through residual analysis, sensitivity studies with respect to collocation density and network width, and comparison with numerical solutions obtained independently using the Haar Wavelet Collocation Method (HWCM). The two approaches show close numerical consistency across the similarity domain. In addition, a parametric study is conducted to investigate the effects of the squeezing parameter, rotation parameter, Soret and Dufour numbers, heat generation parameter, chemical reaction parameter, and hybrid CNT volume fraction on the velocity, temperature, and concentration fields, together with the associated skin-friction coefficient, Nusselt number, and Sherwood number. The results suggest that, under the adopted computational configuration, wavelet-inspired activation functions can provide comparatively more consistent convergence behaviour than conventional activations for the present strongly coupled nonlinear system. The proposed framework offers a mesh-free computational approach for boundary value problems governed by coupled nonlinear differential equations, while the reported observations are specific to the adopted training configuration and are not intended to imply universal performance across all PINN architectures or problem settings.