Neural ordinary differential equations (Neural ODEs) propose the idea that a sequence of layers in a neural network is just a discretisation of an ODE, and thus can instead be directly modelled by a parameterised ODE. This idea has had resounding success in the deep learning literature, with direct or indirect influence in many state of the art ideas, such as diffusion models or time dependant models. Recently, a continuous version of the U-net architecture has been proposed, showing increased performance over its discrete counterpart in many imaging applications and wrapped with theoretical guarantees around its performance and robustness. In this work, we explore the use of Neural ODEs for learned inverse problems, in particular with the well-known Learned Primal Dual algorithm, and apply it to computed tomography (CT) reconstruction.

Abstract-- In this paper, we focus on solving an important class of nonconvex optimization problems which includes many problems for example signal processing over a networked multi-agent system and distributed learning over networks. Motivated by many applications in which the local objective function is the sum of smooth but possibly nonconvex part, and non-smooth but convex part subject to a linear equality constraint, this paper proposes a proximal zeroth-order primal dual algorithm (PZO-PDA) that accounts for the information structure of the problem. This algorithm only utilize the zeroth-order information (i.e., the functional values) of smooth functions, yet the flexibility is achieved for applications that only noisy information of the objective function is accessible, where classical methods cannot be applied. We prove convergence and rate of convergence for PZO-PDA. Numerical experiments are provided to validate the theoretical results.