Deep neural networks have long been criticized for lacking the ability to perform analogical visual reasoning. Here, we propose a neural network model to solve Raven's Progressive Matrices (RPM) - one of the standard intelligence tests in human psychology. Specifically, we design a reasoning block based on the well-known concept of prediction error (PE) in neuroscience. Our reasoning block uses convolution to extract abstract rules from high-level visual features of the 8 context images and generates the features of a predicted answer. PEs are then calculated between the predicted features and those of the 8 candidate answers, and are then passed to the next stage. We further integrate our novel reasoning blocks into a residual network and build a new Predictive Reasoning Network (PredRNet). Extensive experiments show that our proposed PredRNet achieves state-of-the-art average performance on several important RPM benchmarks. PredRNet also shows good generalization abilities in a variety of out-of-distribution scenarios and other visual reasoning tasks. Most importantly, our PredRNet forms low-dimensional representations of abstract rules and minimizes hierarchical prediction errors during model training, supporting the critical role of PE minimization in visual reasoning. Our work highlights the potential of using neuroscience theories to solve abstract visual reasoning problems in artificial intelligence. The code is available at https://github.com/ZjjConan/AVR-PredRNet.