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This paper aims to improve the performance of text-to-SQL parsing by exploring the intrinsic uncertainties in the neural network based approaches (called SUN). From the data uncertainty perspective, it is indisputable that a single SQL can be learned from multiple semantically-equivalent questions.Different from previous methods that are limited to one-to-one mapping, we propose a data uncertainty constraint to explore the underlying complementary semantic information among multiple semantically-equivalent questions (many-to-one) and learn the robust feature representations with reduced spurious associations. In this way, we can reduce the sensitivity of the learned representations and improve the robustness of the parser. From the model uncertainty perspective, there is often structural information (dependence) among the weights of neural networks. To improve the generalizability and stability of neural text-to-SQL parsers, we propose a model uncertainty constraint to refine the query representations by enforcing the output representations of different perturbed encoding networks to be consistent with each other. Extensive experiments on five benchmark datasets demonstrate that our method significantly outperforms strong competitors and achieves new state-of-the-art results. For reproducibility, we release our code and data at https://github.com/AlibabaResearch/DAMO-ConvAI/tree/main/sunsql.