Calibrating chemical kinetics in a reaction-diffusion system is challenging because of complex dynamics governed by tightly coupled chemistry and transport, while experimental observations are often sparse and noisy. We propose a physics consistent diffusion-chemistry coupled neural ordinary differential equation (Diff-Chem Neural ODE) that embeds Arrhenius-structured reaction neurons into a fully differentiable streamline formulation and explicitly accounts for diffusion coupling. This design enables direct gradient-based analysis of kinetic parameters without sampling-based pretraining. We validate this method on burner-stabilized flat and stagnation reacting flows using mechanisms spanning different stiffness ranges. The proposed method reproduces species profiles with near-reference accuracy, whereas a pure chemistry Neural ODE that neglects diffusion coupling may misplace ignition and generate an incorrect thin reaction zone. Diff-Chem Neural ODE is more robust than pure chemistry Neural ODE and provides substantial speedups for gradient evaluation compared with fully discretized computations. In kinetics refinement, optimizing only a limited set of "primal" species reduces the loss by over 98% and simultaneously recovers unobserved variables, demonstrating physically consistent global control. Finally, tests with 1-20% noise in the objective show stable convergence without local overfitting, supporting its applicability under noisy measurements.

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