Enforcing domain constraints through Lagrangian primal-dual learning

Abstract While neural networks have demonstrated remarkable predictive capabilities in various scenarios, they typically struggle to learn to avoid regions of the output space that are considered off-limits due to known domain-specific constraints. This paper presents a novel primal-dual learning approach inspired by augmented Lagrangian methods to address a priori output constraints for neural network predictions. Our solution encodes domain constraints for the output space by using a static Implicit Neural Representation and penalising the violation of these constraints at the loss level. This choice allows full flexibility in incorporating the constraints. We conduct extensive evaluations on several 2D and 3D synthetic datasets with different constraint topologies and two real-world datasets to evaluate the effectiveness of the proposed method. Our approach consistently outperforms methods without constraints in all experiments, yielding higher accuracy and lower constraint violations. Finally, our method shows superior performance improvements in scenarios with limited data availability, opening up potential benefits for various applications. These include geo-localisation tasks, where accurate positioning is crucial, and physical problems with theoretically-imposed constraints. Graphical abstract