What is the relationship between Gu Yanwu and Donglin Academy?

The two are not related.

However, Donglin Academy is related to Gu Xiancheng.

Donglin Academy was founded in 1111, the first year of Zhenghe in the Northern Song Dynasty. It was the place where Yang Shi, the younger brother of the Neo-Confucianists Cheng Hao and Cheng Yi of the Northern Song Dynasty and the famous scholar Yang Shi, gave lectures for a long time. Later abandoned. In the 32nd year of Wanli in the Ming Dynasty, that is, 1604 AD, Donglin scholar Gu Xiancheng and others restored and restored it and gathered people to give lectures here. They advocated the spirit of "reading, lecturing, and patriotism", which aroused widespread response from scholars across the country and became famous for a time. Big move. Donglin Academy became a gathering place for humanities in the Jiangnan region and a major public opinion center for discussing state affairs.

Gu Xiancheng (1550-1612) was a thinker of the Ming Dynasty and leader of the Donglin Party. A native of Wuxi, Jiangsu Province, his courtesy name was Shushi and his alias was Jingyang. He was honored as "Mr. Donglin" because he founded Donglin Academy. In the eighth year of Wanli's reign, he became a Jinshi and was awarded the title of Head of the Household Department. He was in power for more than fifteen years and was relegated to the position of judge of Guiyang Prefecture. He moved to the Ministry of Civil Affairs Yuan Wai Lang and Wen Xuan Lang. In the 22nd year, he disobeyed the imperial decree and dismissed his status. Here he advocated the construction of Donglin Academy with his younger brother Gu Yuncheng, and together with Gao Panlong and others gave lectures in it. They often satirized the government and the government and the public were in harmony. The name of Donglin came from the great master write. Gu Xiancheng, Zhao Nanxing, and Zou Yuan were labeled as the "Three Gentlemen". They were also known as the "Eight Gentlemen of Donglin" together with Gu Yuncheng, Gao Panlong, An Xifan, Liu Yuanzhen, Qian Yiben, Xue Fujiao, and Ye Maocai. They wrote "Gu Duan" Cultural Heritage" etc.

Gu Xiancheng and Gu Yanwu just have the same surname and may be from the same clan.