The recent rise of deep learning has led to numerous applications, including solving partial differential equations using Physics-Informed Neural Networks. This approach has proven highly effective in several academic cases. However, their lack of physical invariances, coupled with other significant weaknesses, such as an inability to handle complex geometries or their lack of generalization capabilities, make them unable to compete with classical numerical solvers in industrial settings. In this work, a limitation regarding the use of automatic differentiation in the context of physics-informed learning is highlighted. A hybrid approach combining physics-informed graph neural networks with numerical kernels from finite elements is introduced. After studying the theoretical properties of our model, we apply it to complex geometries, in two and three dimensions. Our choices are supported by an ablation study, and we evaluate the generalization capacity of the proposed approach.

This study addresses the challenge of multi-objective optimization of a composite shell structure while adhering to constraints on the number of calls to a pseudo-experimental model, simulating real experiments. Two considered objective functions are defined to determine the investigated structure’s dynamic properties and material costs; the optimization involves genetic algorithms, neural surrogate model and multi-fidelity finite-element models. The results of multi-objective optimization were presented as Pareto fronts. A new strategy for preliminary result verification is proposed, significantly reducing the need for a computationally intensive complete verification that requires complex models or experimental investigations. Two different indicators are applied to assess the quality of the obtained Pareto fronts; one is a new one proposed in the paper. Moreover, a multi-fidelity approach is discussed, and three finite element models with different mesh densities are employed, together with a pseudo-experimental model constructed using high-fidelity results and incorporating a nonlinear transformation. However, challenges arise due to the arbitrarily constrained number of pseudo-experiments, limiting future experiments is crucial. The study highlights the need for further analysis of Pareto front indicators and statistical analysis of applied tools like deep neural networks and genetic algorithms. Future research directions include exploring ensemble learning in surrogate models for potential optimization benefits.