Deep learning methods for inverse problems using connections between proximal operators and Hamilton-Jacobi equations

-cross Abstract: Inverse problems are important mathematical problems that seek to recover model parameters from noisy data. Since inverse problems are often ill-posed, they require regularization or incorporation of prior information about the underlying model or unknown variables. Proximal operators, ubiquitous in nonsmooth optimization, are central to this because they encode priors and yield efficient iterative algorithms. They have also recently become key to modern machine learning methods, e.g., plug-and-play methods with learned denoisers and deep neural architectures for learning priors of proximal operators. The latter was developed partly due to recent work characterizing proximal operators of nonconvex priors as subdifferentials of convex potentials. In this work, we propose to leverage connections between proximal operators and Hamilton--Jacobi partial differential equations (HJ PDEs) to develop deep learning architectures for learning the prior. In contrast to other existing methods, we learn the prior directly without recourse to inverting the prior after training. We present numerical results in dimensions up to $64$, where the recovered prior is evaluated in a single forward pass.