Over the past decade, significant progress has been made in theoretical and experimental research in the field of chemical reaction dynamics, moving from triatomic reactions to larger polyatomic reactions. This has challenged the theoretical and computational approaches to polyatomic reaction dynamics in two major areas: the potential energy surface and the dynamics. Highly accurate potential energy surfaces are essential for achieving accurate dynamical information in quantum dynamics calculations. The increased number of degrees of freedom in larger systems poses a significant challenge to the accurate construction of potential energy surfaces. Recently, there has been substantial progress in the development of potential energy surfaces for polyatomic reactive systems. In this article, we review the recent developments made by our group in constructing highly accurately fitted potential energy surfaces for polyatomic reactive systems, based on a neural network approach. A key advantage of the neural network approach is its more faithful representation of the ab initio points. We recently proposed a systematic procedure, based on neural network fitting, for the construction of accurate potential energy surfaces with very small root mean square errors. Based on the neural network approach, we successfully developed potential energy surfaces for polyatomic reactions in the gas phase, including the reactive systems OH_{3}, HOCO, and CH_{5}, and the dissociation of gas-phase molecules on metal surfaces, such as H_{2}O on the Cu(111) surface. These potential energy surfaces were fitted to an unprecedented level of accuracy, representing the most accurate potential energy surfaces calculated for these systems, and were rigorously tested using quantum dynamics calculations. The quantum dynamics calculations based on these potential energy surfaces produce accurate results, which are in good agreement with experiments. We have also proposed a new method for developing permutationally invariant potential energy surfaces, named fundamental-invariant neural networks. Mathematically, fundamental invariants are used to finitely generate the permutation-invariant polynomial ring; thus, fundamental-invariant neural networks can approximate any function to arbitrary accuracy. The use of fundamental invariants minimizes the size of the input permutation-invariant polynomials, which reduces the evaluation time for potential energy calculations, especially for polyatomic systems. Potential energy surfaces for OH_{3} and CH_{4} were constructed using fundamental-invariant neural networks, with their accuracies confirmed by full-dimensional quantum dynamics and bound-state calculations. These developments in the construction of highly accurate potential energy surfaces are expected to extend the theoretical study of reaction dynamics to larger and more complex systems.