Aleatoric uncertainty, or inherent data uncertainty, arises from measurement errors and the inherent variability of real-world phenomena, a topic explored in related research. Conversely, epistemic uncertainty, or model uncertainty, is attributed to incomplete perception of ML models, such as the statistical nature and architectural design of neural networks. Aleatoric uncertainty, typically considered to be irreducible, is challenging to quantify due to difficulties in representation and the absence of redundant data entries in prominent databases. Thus, most research efforts have focused on epistemic uncertainty that can be reduced, such as collecting additional training data in relevant regions of an experimental space. ML models inherently lack mechanisms to directly manage uncertainty, leading to the common practice of employing methods that yield prediction distributions to quantify epistemic uncertainty. Uncertainties inherent in ML models are vital, particularly for applications within materials science where safety and reliability are paramount. For high-stakes fields, effective UQ methods are crucial for informed decision-making and risk assessment. Techniques like active learning, which selectively identifies data points to enhance model training, and reinforcement learning, optimizing actions based on predicted outcomes and their associated risks, rely extensively on UQ to achieve success. In the polymer design process, UQ methods aid in active learning by refining the selection of candidates for the training set, thus enhancing the robustness and stability of predictive models. These models are then leveraged for targeted polymer design, harnessing UQ to streamline the design process and provide interpretability, a critical factor for model validation and verification. Consequently, the integration of UQ methods into ML models emerges as a key driver for innovation and advancement in the targeted design of polymer materials. Recent research in ML has explored various methodologies for UQ, broadly categorized into three main approaches. Monte Carlo-based methods, such as ensemble model and Monte Carlo dropout (MCD), leverage random sampling to estimate predictive distributions, thereby quantifying uncertainty through variability in outcomes. Architecture-specific methods, including evidential deep learning (EDL) and mean-variance estimation (MVE), integrate UQ directly into the model's design, enabling the computation of uncertainty metrics like means and standard deviations alongside predictions. These methods have demonstrated computational efficiency in both the training and evaluation phases, streamlining the process of UQ. Lastly, Bayesian methods, exemplified by Gaussian process regression (GPR) and Bayesian neural network (BNN), adopt a probabilistic modeling approach, using Bayesian inference to derive posterior distributions that inherently capture the model's uncertainty. Each strategy provides a distinct mechanism for assessing uncertainty, from sampling-based estimations to intrinsic probabilistic calculations and architectural integrations, illustrating the diverse tactics employed in ML to address challenges for UQ. These methodologies have been applied to inverse design within polymer field. However, while there have been significant studies focusing on the UQ of inorganic materials, research on polymers, whose complexity presents unique challenges and lacks a systematic approach, such as the representation of molecular structures and design of ML architectures. This gap underscores the critical need for tailored UQ studies that address the nuanced behavior of polymeric materials.