Recent advancements in the integration of artificial intelligence (AI) and machine learning (ML) with physical sciences have led to significant progress in addressing complex phenomena governed by nonlinear partial differential equations (PDEs). This paper explores the application of novel operator learning methodologies to unravel the intricate dynamics of flame instability, particularly focusing on hybrid instabilities arising from the coexistence of Darrieus–Landau (DL) and Diffusive–Thermal (DT) mechanisms. Training datasets encompass a wide range of parameter configurations, enabling the learning of parametric solution advancement operators using techniques such as parametric Fourier Neural Operator (pFNO) and parametric convolutional neural networks (pCNNs). Results demonstrate the efficacy of these methods in accurately predicting short-term and long-term flame evolution across diverse parameter regimes, capturing the characteristic behaviors of pure and blended instabilities. Comparative analyses reveal pFNO as the most accurate model for learning short-term solutions, while all models exhibit robust performance in capturing the nuanced dynamics of flame evolution. This research contributes to the development of robust modeling frameworks for understanding and controlling complex physical processes governed by nonlinear PDEs.