For better clustering performance, appropriate representations are critical. Although many neural network-based metric learning methods have been proposed, they do not directly train neural networks to improve clustering performance. We propose a meta-learning method that train neural networks for obtaining representations such that clustering performance improves when the representations are clustered by the variational Bayesian (VB) inference with an infinite Gaussian mixture model. The proposed method can cluster unseen unlabeled data using knowledge meta-learned with labeled data that are different from the unlabeled data. For the objective function, we propose a continuous approximation of the adjusted Rand index (ARI), by which we can evaluate the clustering performance from soft clustering assignments. Since the approximated ARI and the VB inference procedure are differentiable, we can backpropagate the objective function through the VB inference procedure to train the neural networks. With experiments using text and image data sets, we demonstrate that our proposed method has a higher adjusted Rand index than existing methods do.