Data-Driven Numerical Method Construction Via Physics-Informed Hard-Constrained Manifold Learning: Finite Elements as a Demonstration
Abstract
Can data-driven learning move beyond approximating solutions of the prescribed PDE problems to constructing reusable numerical methods? We investigate this question through finite elements and develop a physics-informed hard-constrained manifold learning Finite Element (HCML-FE) framework. Physics-informed linear hard constraints on the element stiffness matrix are analytically transformed into algebraic relations among its entries. Symbolic Gaussian elimination then identifies the independent stiffness parameters and deterministic reconstruction operator. A neural-network predicts only these independent parameters, while the complete stiffness matrix is reconstructed analytically so that the prescribed linear mechanical constraints are satisfied exactly. For the four-node plane-stress and eight-node solid-shell elements considered here, only 3 and 78 independent stiffness parameters are required, respectively, to reconstruct the stiffness matrices. Their numbers and matrix locations remain unchanged over the admissible ranges of element geometry and material parameters. Reference data are obtained from independently developed finite-element programs and commercial finite-element software, allowing the framework to be examined using element technologies of different origins. Numerical results show that the HCML-FE passes rigid-body zero-energy and prescribed patch tests, reproduces characteristic element behaviors under locking, mesh distortion, and mesh refinement, and…
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