Diffusion models have quickly become the state-of-the-art for generation tasks across many different data modalities. An important ability of diffusion models is the ability to encode samples from the data distribution back into the sampling prior distribution. This is useful for performing alterations to real data samples along with guided generation via the continuous adjoint equations. We propose an algebraically reversible solver for diffusion SDEs that can exactly invert real data samples into the prior distribution. Diffusion models have quickly become the state-of-the-art in many different modalities in generation, e.g., audio (Liu et al., 2023), images (Rombach et al., 2022), video (Blattmann et al., 2023), protein generation (Skreta et al., 2024), & c.

Neural ODEs are a recently developed model class that combine the strong model priors of differential equations with the high-capacity function approximation of neural networks. One advantage of Neural ODEs is the potential for memory-efficient training via the continuous adjoint method. However, memory-efficient training comes at the cost of approximate gradients. Therefore, in practice, gradients are often obtained by simply backpropagating through the internal operations of the forward ODE solve - incurring high memory cost. Interestingly, it is possible to construct algebraically reversible ODE solvers that allow for both exact gradients and the memory-efficiency of the continuous adjoint method. Unfortunately, current reversible solvers are low-order and suffer from poor numerical stability. The use of these methods in practice is therefore limited. In this work, we present a class of algebraically reversible solvers that are both high-order and numerically stable. Moreover, any explicit numerical scheme can be made reversible by our method. This construction naturally extends to numerical schemes for Neural CDEs and SDEs.