Title: Quadratic models for understanding neural network dynamics. Abstract: In this work, we show that recently proposed quadratic models capture optimization and generalization properties of wide neural networks that cannot be captured by linear models.  In particular, we prove that quadratic models for shallow ReLU networks exhibit the "catapult phase" from Lewkowycz et al. (2020) that arises when training such models with large learning rates. We then empirically show that the behaviour of quadratic models parallels that of neural networks in generalization, especially in the catapult phase regime. Our analysis further demonstrates that quadratic models are an effective tool for analysis of neural networks. 