Unrolling operator splitting in learning PDEs for object detection

This is the authors' abstract. We don't add key points for this paper.

Object detection presents significant challenges due to the variability in object scale, location, and orientation within images. Most state-of-the-art detectors are based on convolutional or Transformer architectures, which, while effective, often result in deep, opaque models that generalise poorly and lack interpretability. In contrast, iterative algorithms offer greater transparency and generalisation, albeit at the cost of efficiency and accuracy. In this work, we reformulate object detection as a partial differential equation (PDE)-constrained optimal control problem. This formulation exploits linear combinations of fundamental differential invariants—such as translation and rotation invariance—to embed structural priors into the learning process. We solve this problem using operator splitting via the Alternating Direction Method of Multipliers (ADMM), and unroll each optimisation step into a network layer, yielding a novel architecture: ADMM-ODNet. This approach provides a principled and interpretable alternative to conventional deep networks. Experimental results on the Corel, Pascal VOC and COCO datasets demonstrate that ADMM-ODNet outperforms leading models such as Cascade Mask R-CNN, Swin Transformer, Deformable DETR, DINO, DN-and RT-DETR, and achieves performance comparable to Plain DETR and YOLO, while requiring significantly fewer parameters. Differential invariant of an attribute F that is closed under total differentiation with respect to the differential constraints. The total derivative F ′ points inwards along the differential constraints or transversally to F without pointing outwards to ¬ F , as indicated by the arrows within the constraints. • Reformulate object detection as a PDE-constrained optimal control problem. • Propose an optimisation scheme based on operator splitting (ADMM). • Unroll ADMM sequence iterations into network layers of ADMM-ODNet. • A lightweight detection head for efficient bounding box generation and classification. • Extensive experiments demonstrate effective object detection performance.