The realization of extremely low power compact devices is critical for ubiquitous computing. In that context, approximate computing is one approach towards improving computing efficiency in error-tolerant applications that may endure a small loss in accuracy in exchange for better efficiency. In this paper, we present a framework for automated synthesis of crossbar designs that implement approximate versions of compute kernels using flow-based computing. We propose Approximate Free Binary Decision Diagrams (AFBDDs) using a relaxed function equality criterion in node merging. We create AFBDD representations of kernels which are then mapped onto crossbars. These crossbars can compute an approximation of the original compute kernel. Our framework allows us to easily tune the accuracy-efficiency trade-off for an approximate design. We generate designs for RevLib and MCNC benchmarks, and basic image processing kernels. Our designs are up to 80% more compact than exact designs while retaining ≈ 90% accuracy.