Title: Regularization Neural Networks via Constrained Virtual  Movement Field. Abstract: We provide a novel thinking of regularization neural networks. We smooth the objective of neural networks w.r.t small adversarial perturbations of the inputs. Different from previous works, we assume the adversarial perturbations are caused by the movement field. When the magnitude of movement field approaches 0, we call it virtual movement field. By introducing the movement field, we cast the problem of finding adversarial perturbations into the problem of finding adversarial movement field. By adding proper geometrical constraints to the movement field, such smoothness can be approximated in closed-form by solving a min-max problem and its geometric meaning is clear. We define the approximated smoothness as the regularization term.  We derive three regularization terms as running examples which measure the smoothness w.r.t shift, rotation and scale respectively by adding different constraints. We evaluate our methods on synthetic data, MNIST and CIFAR-10. Experimental results show that our proposed method can significantly improve the baseline neural networks. Compared with the state of the art regularization methods, proposed method achieves a tradeoff between accuracy and geometrical interpretability as well as computational cost.