One fundamental task for NLP is to determine the similarity between two texts and evaluate the extent of their likeness. The previous methods for the Persian language have low accuracy and are unable to comprehend the structure and meaning of texts effectively. Additionally, these methods primarily focus on formal texts, but in real-world applications of text processing, there is a need for robust methods that can handle colloquial texts. This requires algorithms that consider the structure and significance of words based on context, rather than just the frequency of words. The lack of a proper dataset for this task in the Persian language makes it important to develop such algorithms and construct a dataset for Persian text. This paper introduces a new transformer-based model to measure semantic similarity between Persian informal short texts from social networks. In addition, a Persian dataset named FarSSiM has been constructed for this purpose, using real data from social networks and manually annotated and verified by a linguistic expert team. The proposed model involves training a large language model using the BERT architecture from scratch. This model, called FarSSiBERT, is pre-trained on approximately 104 million Persian informal short texts from social networks, making it one of a kind in the Persian language. Moreover, a novel specialized informal language tokenizer is provided that not only performs tokenization on formal texts well but also accurately identifies tokens that other Persian tokenizers are unable to recognize. It has been demonstrated that our proposed model outperforms ParsBERT, laBSE, and multilingual BERT in the Pearson and Spearman's coefficient criteria. Additionally, the pre-trained large language model has great potential for use in other NLP tasks on colloquial text and as a tokenizer for less-known informal words.