differentiable projection layer

This work addresses the challenge of integrating hard constraints in deep learning, where conventional orthogonal projections onto constraint sets often induce gradient saturation and impede optimization. To overcome this limitation, the authors propose a differentiable soft radial projection layer that maps inputs from Euclidean space radially into the interior of the feasible set, thereby guaranteeing strict feasibility while avoiding vanishing gradients. The method features a Jacobian matrix that is full-rank almost everywhere, effectively circumventing the gradient degeneracy associated with traditional boundary-based projections, and preserves the universal approximation capability of neural networks. By combining differentiable reparameterization with end-to-end training, the proposed approach consistently outperforms state-of-the-art optimization and projection baselines in both convergence speed and solution quality.

This work addresses the challenge of unphysical and unstable simulations in existing learning-based fluid dynamics models, which often fail to strictly satisfy the incompressible continuity equation due to the absence of hard physical constraints. The authors propose a unified framework that enforces incompressibility by hard-coding the divergence-free constraint, enabling both deterministic and generative models to operate exclusively within the solenoidal subspace. The key innovation lies in the first-time integration of a differentiable spectral Leray projector—derived from Helmholtz–Hodge decomposition—with a curl-driven Gaussian reference measure, ensuring that both model outputs and generated distributions rigorously adhere to zero divergence. Experiments on the two-dimensional Navier–Stokes equations demonstrate exact incompressibility up to discretization error, significantly enhancing long-term simulation stability and physical fidelity.

In real-time robotic safety control—particularly for control barrier functions—accurate, differentiable computation of distances and their derivatives between objects and environments is essential. However, the Euclidean distance is non-differentiable at contact points, while existing differentiable distance methods suffer from complex analytical forms, poor adaptability to convex polyhedral geometries, and failure to converge to zero under object overlap. Method: This paper proposes a novel differentiable distance metric grounded in a generalized alternating projection framework. We derive a compact, closed-form smooth projection operator for general convex polyhedra, ensuring global differentiability and exact distance convergence to zero upon overlap. Contribution/Results: The method is implemented in the Python simulation platform UAIBot. Experiments demonstrate high computational efficiency, numerically stable gradients, and significantly improved stability and practicality of real-time safety-critical control.

This work addresses the limitation of existing diffusion models in generating incompressible flow fields, which often neglect physical constraints or enforce them only through soft penalties. To overcome this, we propose a diffusion-based generative framework that integrates hard geometric constraints with soft physical regularization. Our approach combines boundary-condition-guided diffusion, a physics-informed loss incorporating divergence penalties, and a projection-constrained reverse sampling scheme based on a geometry-aware Helmholtz–Hodge decomposition. Furthermore, we bridge the generative model with the intrinsic geometry of incompressible flows via constrained Langevin dynamics on manifolds. Experiments demonstrate that our method significantly outperforms baseline approaches on both analytical Navier–Stokes solutions and complex obstacle scenarios, achieving substantial improvements in divergence error, spectral accuracy, vorticity statistics, and boundary consistency.

This work addresses directed acyclic graph (DAG) structure learning from observational data, tackling two core challenges in differentiable DAG optimization: gradient vanishing and the difficulty of enforcing the DAG constraint via differentiable formulations. We establish, for the first time, a theoretical connection between analytic function families and the DAG constraint, proposing a closed class of analytic functions—specifically, positive-coefficient convergent power series—that supports differentiation, addition, and multiplication, thereby enabling fully differentiable DAG regularization. Integrating graph-structured priors with an efficient automatic differentiation scheme, our approach ensures stable gradient propagation and exact DAG constraint satisfaction. Extensive experiments across diverse benchmark tasks demonstrate significant improvements over existing state-of-the-art methods, with superior constraint fidelity, computational efficiency, and robustness. The implementation is publicly available.

This study addresses the PDE-constrained inverse problem of correcting Reynolds-averaged Navier–Stokes (RANS) turbulence models by introducing an end-to-end differentiable framework. The approach embeds a PDE solver into PyTorch’s automatic differentiation graph via implicit layers and incorporates a trainable additive correction term, enabling joint optimization of model parameters or neural networks. Built upon the BROADCAST solver, this work presents a unified and user-friendly differentiable PDE interface and, for the first time, achieves end-to-end differentiable correction for compressible-flow RANS models. The framework successfully optimizes the production-term coefficient in the Spalart–Allmaras model and reconstructs the eddy viscosity field in two canonical cases—NASA’s wall-mounted hump and the VKI LS-59 turbine blade—demonstrating its effectiveness and flexibility for turbulence modeling and broader physics-informed PDE inverse problems.