Proximal Splitting Methods for Hybrid Differentiable Models

Operator splitting methods are at the foundation of many numerical solvers for partial differential equations. In parallel, unrolled and hybrid learning-based architectures have been introduced to enhance classical solvers, but their design is rarely linked to the underlying problem structure. In this work, we propose a unifying framework that explicitly links operator splitting algorithms from optimization with unrolled hybrid architectures. We show that each operator splitting scheme naturally defines an unrolled architecture, which recovers a wide range of existing plug-and-play and hybrid models as special cases. Using this framework, we design new unrolled hybrid architectures and validate them on benchmark fluid dynamics simulations, where they achieve improved accuracy and stability.

In this work, we show that a broad family of mathematical optimization methods, known as operator splitting algorithms, can serve as a blueprint for designing hybrid machine learning architectures. Our framework reveals that many existing approaches are actually special cases of the same underlying idea. Building on this insight, we develop new hybrid architectures that more closely follow the structure of the original mathematical problem. We test these methods on fluid dynamics simulations, which are widely used in science and engineering. The results show that our approach produces more accurate and stable predictions than existing methods. By creating a clearer link between established numerical algorithms and modern machine learning techniques, this work provides a systematic way to build more reliable AI-assisted scientific computing tools.