As a representative next-generation device/circuit technology beyond CMOS, diffractive optical neural networks (DONNs) have shown promising advantages over conventional deep neural networks due to extreme fast computation speed (light speed) and low energy consumption. However, there is a mismatch, i.e., significant prediction accuracy loss, between the DONN numerical modelling and physical optical device deployment, because of the interpixel interaction within the diffractive layers. In this work, we propose a physics-aware diffractive optical neural network training framework to reduce the performance difference between numerical modeling and practical deployment. Specifically, we propose the roughness modeling regularization in the training process and integrate the physics-aware sparsification method to introduce sparsity to the phase masks to reduce sharp phase changes between adjacent pixels in diffractive layers. We further develop $2π$ periodic optimization to reduce the roughness of the phase masks to preserve the performance of DONN. Experiment results demonstrate that, compared to state-of-the-arts, our physics-aware optimization can provide $35.7\%$, $34.2\%$, $28.1\%$, and $27.3\%$ reduction in roughness with only accuracy loss on MNIST, FMNIST, KMNIST, and EMNIST, respectively.