Many real-world engineering problems require simulations to evaluate the design objectives and constraints. Often, due to the complexity of the system model, simulations can be prohibitive in terms of computation time. One approach to overcome this issue is to construct a surrogate model, which approximates the original model. The focus of this work is on the data-driven surrogate models, in which empirical approximations of the output are performed given the input parameters. Recently neural networks (NN) have re-emerged as a popular method for constructing data-driven surrogate models. Although, NNs have achieved excellent accuracy and are widely used, they pose their own challenges. This work addresses two common challenges, the need for: (1) hardware acceleration and (2) uncertainty quantification (UQ) in the presence of input variability. The high demand in the inference phase of deep NNs in cloud servers/edge devices calls for the design of low power custom hardware accelerators. The first part of this work describes the design of an energy-efficient long short-term memory (LSTM) accelerator. The overarching goal is to aggressively reduce the power consumption and area of the LSTM components using approximate computing, and then use architectural level techniques to boost the performance. The proposed design is synthesized and placed and routed as an application-specific integrated circuit (ASIC). The results demonstrate that this accelerator is 1.2X and 3.6X more energy-efficient and area-efficient than the baseline LSTM. In the second part of this work, a robust framework is developed based on an alternate data-driven surrogate model referred to as polynomial chaos expansion (PCE) for addressing UQ. In contrast to many existing approaches, no assumptions are made on the elements of the function space and UQ is a function of the expansion coefficients. Moreover, the sensitivity of the output with respect to any subset of the input variables can be computed analytically by post-processing the PCE coefficients. This provides a systematic and incremental method to pruning or changing the order of the model. This framework is evaluated on several real-world applications from different domains and is extended for classification tasks as well.