Physics-informed meta-learning for elasticity problems with geometric parameterization

In this paper, Model-Agnostic Meta-Learning based on Deep Energy Method (MAML-DEM), a novel meta-learning framework is developed for the geometric parameterization. A single meta-trained neural network efficiently solves diverse 2D linear elasticity problems in plates with complex and varying topologies, including those containing multiple mixed circular and elliptical holes. Conceptual studies further demonstrate the framework's potential to generalize to non-uniform boundary conditions and more complex L-shaped hole geometries. By leveraging the variational principle of minimum potential energy, the model avoids the unstable gradients linked to second-order derivatives in standard Physics-Informed Neural Networks (PINNs). Additionally, a geometry-aware adaptive sampling method is employed to capture high-stress areas around geometric discontinuities precisely. During meta-training, the model learns a broad physical understanding applicable across various tasks. Results show that this approach can adapt very quickly to new and unseen geometries, achieving speeds up to 69x faster than training a specific model from scratch. The MAML-DEM framework exhibits superior accuracy and stability over conventional PINN methods, while also demonstrating strong generalization capability to tasks beyond its training data, effectively handling variations in topology, boundary conditions, and geometric complexity. This work highlights the potential of meta-learning to transform physics-informed simulations into practical and efficient tools for rapid engineering design and analysis.