Studies on artificial neural network have been conducted for a long time, and its contribution has been shown in many fields. However, the application of neural networks in the real world domain is still a challenge, since nature does not always provide the required satisfactory conditions. One example is the class size unbalanced condition in which one class is heavily under-represented compared to another class. This condition is often found in the real world domain and presents several difficulties for algorithms that assume the balanced condition of the classes. In this paper, we propose a method for solving problems posed by imbalanced training sets by applying the modified large-scale neural network "CombNET-II." CombNET-II consists of two types of neural networks. The first type is a one-layer vector quantization neural network to turn the problem into a more balanced condition. The second type consists of several modules of three-layered multilayer perceptron trained by backpropagation for finer classification. CombNET-II combines the two types of neural networks to solve the problem effectively within a reasonable time. The performance is then evaluated by turning the model into a practical application for a fog forecasting problem. Fog forecasting is an imbalanced training sets problem, since the probability of fog appearance in the observation location is very low. Fog events should be predicted every 30 minutes based on the observation of meteorological conditions. Our experiments showed that CombNET-II could achieve a high prediction rate compared to the k-nearest neighbor classifier and the three-layered multilayer perceptron trained with BP. Part of this research was presented in the 1999 Fog Forecasting Contest sponsored by Neurocomputing Technical Croup of IEICE, Japan; and CombNET-II achieved the highest accuracy among the participants.