DiffSlack: Learning under Nonlinear Inequality Constraints via Learnable Slack Variables

Enforcing nonlinear inequality constraints in neural networks remains challenging, especially when the output is subject to many coupled constraints. Existing hard constraint methods often impose structural restrictions on the constraint set or introduce substantial computational overhead for large-scale nonlinear problems. Here, we propose DiffSlack, a differentiable projection layer for nonlinear inequality-constrained neural prediction. DiffSlack reformulates inequalities as equalities with learnable slack variables, which are predicted as part of the augmented network output and provide a data-driven warm start for damped Gauss-Newton projection. The projection layer maps raw predictions onto the augmented feasible manifold while preserving end-to-end differentiability. A two-stage curriculum further stabilizes training and improves constraint satisfaction. We evaluate DiffSlack on vehicle path planning with 200 nonlinear inequality constraints from collision avoidance, curvature limits, and waypoint spacing. Compared with existing learning-based baselines, DiffSlack achieves a higher planning success rate and stronger geometric constraint satisfaction under a comparable inference budget. Ablation studies further show that the hard projection layer reduces sensitivity to supervision quality. Closed-loop tracking in CARLA and real-world vehicle experiments confirms the executability of the generated trajectories. These results demonstrate that DiffSlack provides a practical and scalable approach to embedding hard inequality constraints into neural networks for engineering applications.

Self-Supervised Path Planning in Unstructured Environments via Global-Guided Differentiable Hard Constraint Projection

Deploying deep learning agents for autonomous navigation in unstructured environments faces critical challenges regarding safety, data scarcity, and limited computational resources. Traditional solvers often suffer from high latency, while emerging learning-based approaches struggle to ensure deterministic feasibility. To bridge the gap from embodied to embedded intelligence, we propose a self-supervised framework incorporating a differentiable hard constraint projection layer for runtime assurance. To mitigate data scarcity, we construct a Global-Guided Artificial Potential Field (G-APF), which provides dense supervision signals without manual labeling. To enforce actuator limitations and geometric constraints efficiently, we employ an adaptive neural projection layer, which iteratively rectifies the coarse network output onto the feasible manifold. Extensive benchmarks on a test set of 20,000 scenarios demonstrate an 88.75\% success rate, substantiating the enhanced operational safety. Closed-loop experiments in CARLA further validate the physical realizability of the planned paths under dynamic constraints. Furthermore, deployment verification on an NVIDIA Jetson Orin NX confirms an inference latency of 94 ms, showing real-time feasibility on resource-constrained embedded hardware. This framework offers a generalized paradigm for embedding physical laws into neural architectures, providing a viable direction for solving constrained optimization in mechatronics. Source code is available at: https://github.com/wzq-13/SSHC.git.