Hybrid Neural Computing Frameworks for Nonlinear and Time-Dependent Diffusive Flow Solutions.

The present paper proposes a hybrid neural computing framework (LMBNNs) which has combined artificial neural networks with Levenberg-Marquardt back-propagation by applying supervised learning to resolve nonlinear, time-dependent diffusion flows presented by Burgers equation. The framework overcomes difficulties in simulation of complex fluid dynamics in which conventional techniques fail due to the presence of shock phenomena, bifurcation effects and inability to implement effective numerical approaches. Our physics-informed design embeds conservation laws directly into the learning process while adaptive optimization methods automatically adjust to solution characteristics, ensuring both accuracy and stability .The approach proves to be superior to traditional methods in terms of better predictability of flow transition and major improvements in computational performance in different regimes of viscosity. LMBNNs through rigorous validation demonstrate strong handling of time-dependent solutions and accurate identification of bifurcation point becoming key elements of taking into account interfacial dynamics in both the standard neural networks and numerical methods. This is possible due to the dynamic regularization that ensures the stability of the solution in addition to accommodation of industrial-scale problems that may vary in viscosity due to the scalability of the framework. Important innovations are an intelligent damping mechanism of ill-conditioned systems and a novel method of shock-front localization. Among current applications span microfluidics, multiphase flows, and other engineering systems where precise simulation of the nonlinear transport phenomenon is applicable. LMBNNs create a new paradigm for resolving complex PDEs that combines the pattern recognition strengths of machine learning with the precision of numerical analysis, offering transformative potential for both scientific computing and industrial applications. The generalization ability of the framework is validated with the help of extensive stability analysis which proves it to be a powerful instrument that can open the way to new possibilities of computational fluid dynamics and related academic disciplines.