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Balancing the trade-off between safety and efficiency is of significant importance for path planning under uncertainty. Many risk-aware path planners have been developed to explicitly limit the probability of collision to an acceptable bound in uncertain environments. However, convex obstacles or Gaussian uncertainties are usually assumed to make the problem tractable in the existing method. These assumptions limit the generalization and application of path planners in real-world implementations. In this article, we propose to apply deep learning methods to the sampling-based planner, developing a novel risk bounded near-optimal path planning algorithm named neural risk-aware RRT (NR-RRT). Specifically, a deterministic risk contours map is maintained by perceiving the probabilistic nonconvex obstacles, and a neural network sampler is proposed to predict the next most-promising safe state. Furthermore, the recursive divide-and-conquer planning and bidirectional search strategies are used to accelerate the convergence to a near-optimal solution with guaranteed bounded risk. Worst-case theoretical guarantees can also be proven owing to a standby safety guaranteed planner utilizing a uniform sampling distribution. Simulation experiments demonstrate that the proposed algorithm outperforms the state-of-the-art remarkably for finding risk bounded low-cost paths in seen and unseen environments with uncertainty and nonconvex constraints.