FlowGrad: Controlling the Output of Generative ODEs With Gradients
Authors: Xingchao Liu, Lemeng Wu, Shujian Zhang, Chengyue Gong, Wei Ping, Qiang Liu

Generative modeling with ordinary differential equations (ODEs) has achieved fantastic results on a variety of applications. Yet, few works have focused on controlling the generated content of a pre-trained ODE-based generative model. In this paper, we propose to optimize the output of ODE models according to a guidance function to achieve controllable generation. We point out that, the gradients can be efficiently back-propagated from the output to any intermediate time steps on the ODE trajectory, by decomposing the back-propagation and computing vector-Jacobian products. To further accelerate the computation of the back-propagation, we propose to use a non-uniform discretization to approximate the ODE trajectory, where we measure how straight the trajectory is and gather the straight parts into one discretization step. This allows us to save  90% of the back-propagation time with ignorable error. Our framework, named FlowGrad, outperforms the state-of-the-art baselines on text-guided image manipulation. Moreover, FlowGrad enables us to find global semantic directions in frozen ODE-based generative models that can be used to manipulate new images without extra optimization.