2025/05/19 by Tianju Xue, Xue, Tianju
Computer Science · Engineering · Physics and Astronomy · #Advanced Multi-Objective Optimization Algorithms #Computational Engineering #FOS: Computer and information sciences #Finance #Model Reduction and Neural Networks #Topology Optimization in Engineering #and Science (cs.CE)
paper · pdf · doi:10.48550/arxiv.2505.12646
openalex publication_date 2025/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Differentiable programming is revolutionizing computational science by enabling automatic differentiation (AD) of numerical simulations. While first-order gradients are well-established, second-order derivatives (Hessians) for implicit functions in finite-element-based differentiable physics remain underexplored. This work bridges this gap by deriving and implementing a framework for implicit Hessian computation in PDE-constrained optimization problems. We leverage primitive AD tools (Jacobian-vector product/vector-Jacobian product) to build an algorithm for Hessian-vector products and validate the accuracy against finite difference approximations. Four benchmarks spanning linear/nonlinear, 2D/3D, and single/coupled-variable problems demonstrate the utility of second-order information. Results show that the Newton-CG method with exact Hessians accelerates convergence for nonlinear inverse problems (e.g., traction force identification, shape optimization), while the L-BFGS-B method suffices for linear cases. Our work provides a robust foundation for integrating second-order implicit differentiation into differentiable physics engines, enabling faster and more reliable optimization.