Implicit differentiation with second-order derivatives and benchmarks in finite-element-based differentiable physics
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915291941830656 |
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| author | Xue, Tianju |
| author_facet | Xue, Tianju |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_12646 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Implicit differentiation with second-order derivatives and benchmarks in finite-element-based differentiable physics Xue, Tianju Computational Engineering, Finance, and Science 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. |
| title | Implicit differentiation with second-order derivatives and benchmarks in finite-element-based differentiable physics |
| topic | Computational Engineering, Finance, and Science |
| url | https://arxiv.org/abs/2505.12646 |