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Processing-in-memory (PIM) architectures have demonstrated great potential in accelerating numerous deep learning tasks. Particularly, resistive random-access memory (RRAM) devices provide a promising hardware substrate to build PIM accelerators due to their abilities to realize efficient in-situ vector-matrix multiplications (VMMs). However, existing PIM accelerators suffer from frequent and energy-intensive analog-to-digital (A/D) conversions, severely limiting their performance. This paper presents a new PIM architecture to efficiently accelerate deep learning tasks by minimizing the required A/D conversions with analog accumulation and neural approximated peripheral circuits. We first characterize the different dataflows employed by existing PIM accelerators, based on which a new dataflow is proposed to remarkably reduce the required A/D conversions for VMMs by extending shift and add (S+A) operations into the analog domain before the final quantizations. We then leverage a neural approximation method to design both analog accumulation circuits (S+A) and quantization circuits (ADCs) with RRAM crossbar arrays in a highly-efficient manner. Finally, we apply them to build an RRAM-based PIM accelerator (i.e., \textbf{Neural-PIM}) upon the proposed analog dataflow and evaluate its system-level performance. Evaluations on different benchmarks demonstrate that Neural-PIM can improve energy efficiency by 5.36x (1.73x) and speed up throughput by 3.43x (1.59x) without losing accuracy, compared to the state-of-the-art RRAM-based PIM accelerators, i.e., ISAAC (CASCADE).