This paper proposes a method to optimize communication code rates via the application of neural polar decoders (NPDs). Employing this approach enables simultaneous optimization of code rates over input distributions while providing a practical coding scheme within the framework of polar codes. The proposed approach is designed for scenarios where the channel model is unknown, treating the channel as a black box that produces output samples from input samples. We employ polar codes to achieve our objectives, using NPDs to estimate mutual information (MI) between the channel inputs and outputs, and optimize a parametric model of the input distribution. The methodology involves a two-phase process: a training phase and an inference phase. In the training phase, two steps are repeated interchangeably. First, the estimation step estimates the MI of the channel inputs and outputs via NPDs. Second, the improvement step optimizes the input distribution parameters to maximize the MI estimate obtained by the NPDs. In the inference phase, the optimized model is used to construct polar codes. This involves incorporating the Honda-Yamamoto (HY) scheme to accommodate the optimized input distributions and list decoding to enhance decoding performance. Experimental results on memoryless and finite-state channels (FSCs) demonstrate the effectiveness of our approach, particularly in cases where the channel's capacity-achieving input distribution is non-uniform. For these cases, we show significant improvements in MI and bit error rates (BERs) over those achieved by uniform and independent and identically distributed (i.i.d.) input distributions, validating our method for block lengths up to 1024. This scalable approach has potential applications in real-world communication systems, bridging theoretical capacity estimation and practical coding performance.