This paper presents a neural network approach for the identification of gamma emitting radionuclides measured by Silicon Photomultipliers (SiPMs). SiPMs have wide-ranging applications in the field of radiation monitoring and medical imaging due to their high quantum efficiency, high gain, and compactness. For better accuracy on the measured information, however, we should consider the disadvantages: dark count rate, optical crosstalk, and temperature sensitivity. Regarding temperature dependencies, conventional approaches are mainly focused on compensating gain variances against temperature by adding circuits or calibrating signals. In contrast with previous works, we propose a new approach exploiting two-layer fully connected neural network with a SiPM. The neural network is composed of rectified linear unit (ReLU) layers and a softmax layer. A Saint-Gobain Lutetium-yttrium oxyorthosilicate (LYSO) and a SensL MicroFJ SiPM combined with a charge sensitive amplifier (CSA) circuit were used in the data acquisition to detect Cs-137 and Eu-152. The decreasing logistic regression cost function shows that the proposed neural network converges to a global minimum, verifying the possibility of distinguishing Cesium from Europium in the mixed radioactive environment.