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Rapid advances in artificial intelligence (AI) technology have led to significant accuracy improvements in a myriad of application domains at the cost of larger and more compute-intensive models. Training such models on massive amounts of data typically requires scaling to many compute nodes and relies heavily on collective communication algorithms, such as all-reduce, to exchange the weight gradients between different nodes. The overhead of these collective communication operations in a distributed AI training system can bottleneck its performance, with more pronounced effects as the number of nodes increases. In this paper, we first characterize the all-reduce operation overhead by profiling distributed AI training. Then, we propose a new smart network interface card (NIC) for distributed AI training systems using field-programmable gate arrays (FPGAs) to accelerate all-reduce operations and optimize network bandwidth utilization via data compression. The AI smart NIC frees up the system's compute resources to perform the more compute-intensive tensor operations and increases the overall node-to-node communication efficiency. We perform real measurements on a prototype distributed AI training system comprised of 6 compute nodes to evaluate the performance gains of our proposed FPGA-based AI smart NIC compared to a baseline system with regular NICs. We also use these measurements to validate an analytical model that we formulate to predict performance when scaling to larger systems. Our proposed FPGA-based AI smart NIC enhances overall training performance by 1.6x at 6 nodes, with an estimated 2.5x performance improvement at 32 nodes, compared to the baseline system using conventional NICs.