According to one aspect, a method to estimate a state of a battery includes receiving physical quantity information about a sensed physical quantity of a battery, obtaining estimated information of the battery from a battery model based on the received physical quantity information. The battery model includes a training model configured to determine internal battery information comprising potential information of an internal material of the battery based on the physical quantity information and a mathematical function. The method further includes calculating an ion concentration in the battery using the mathematical function based on the internal battery information to determine the estimated information of the battery.