Potential energy surfaces are fundamental to materials simulation, providing acomprehensive landscape of the energy variations as a function of atomic positions within a system. These surfaces are crucial for understanding the behavior of materials at the atomic and molecular levels. By mapping out the potential energy surface, researchers can predict stable configurations, transition states, and thermodynamic and kinetic observables, essential for the design of novel materials. However, constructing accurate potential energy surfaces presents significant challenges due to the complexity constructing a potential with nearly the same accuracy of quantum mechanical calculations, the transferability to explore the high-dimensional space of atomic configurations, and the efficiency to be evaluated without the need for extensive computational resources. In this work, group theoretical irreducible derivativemethods are used to construct accurate Taylor series expansions of the potential energy surface. A new condition number optimized basis is developed which guarantees no amplification of error at the smallest computational cost allowed by group theory. Using this highly efficient method, the physics of the metal-insulator transition in LuNiO₃ is investigated within both a purely harmonic model and simplified anharmonic model, yielding reasonable and good agreement with the experimental phase transition temperature, respectively. Then the irreducible derivatives approach is used to benchmark the phonon anharmonicity of the several popular machine learning potentials: Gaussian approximation potentials, artificial neural network potentials, and graph neural network potentials. The benchmark indicated that the graph neural network potentials had the ability to accurate reproduce quantum mechanical derivatives up to 5th-order. Finally, insights from the benchmark are used to train an accurate graph neural network potential for strongly correlated UO₂ which yielded excellent agreement with quantum mechanical calculations and experiment. These methodological advancements underscore the potential of combining grouptheoretical approaches with cutting-edge machine learning techniques to enhance the precision and efficiency of materials simulations. By achieving high accuracy in modeling complex phenomena such as phase transitions and phonon anharmonicity, this work paves the way for future studies to not only explore the design of novel materials with unprecedented properties, but to design new machine learning techniques where group theory is built from the ground up.