Deep learning is a promising approach to early DRV (Design Rule Violation) prediction. However, non-deterministic parallel routing hampers model training and degrades prediction accuracy. In this work, we propose a stochastic approach, called LGC-Net, to solve this problem. In this approach, we develop new techniques of Gaussian random field layer and focal likelihood loss function to seamlessly integrate Log Gaussian Cox process with deep learning. This approach provides not only statistical regression results but also classification ones with different thresholds without retraining. Experimental results with noisy training data on industrial designs demonstrate that LGC-Net achieves significantly better accuracy of DRV density prediction than prior arts.