To develop faster solvers for governing physical equations in solid
mechanics, we introduce a method that parametrically learns the solution to
mechanical equilibrium. The introduced method outperforms traditional ones in
terms of computational cost while acceptably maintaining accuracy. Moreover, it
generalizes and enhances the standard physics-informed neural networks to learn
a parametric solution with rather sharp discontinuities. We focus on
micromechanics as an example, where the knowledge of the micro-mechanical
solution, i.e., deformation and stress fields for a given heterogeneous
microstructure, is crucial. The parameter under investigation is the Young
modulus distribution within the heterogeneous solid system. Our method,
inspired by operator learning and the finite element method, demonstrates the
ability to train without relying on data from other numerical solvers. Instead,
we leverage ideas from the finite element approach to efficiently set up loss
functions algebraically, particularly based on the discretized weak form of the
governing equations. Notably, our investigations reveal that physics-based
training yields higher accuracy compared to purely data-driven approaches for
unseen microstructures. In essence, this method achieves independence from data
and enhances accuracy for predictions beyond the training range. The
aforementioned observations apply here to heterogeneous elastic
microstructures. Comparisons are also made with other well-known operator
learning algorithms, such as DeepOnet, to further emphasize the advantages of
the newly proposed architecture.

为了开发更快速解决固体力学中的物理方程的求解器，我们引入了一种参数化学习力学平衡解的方法，该方法在计算成本方面优于传统方法，同时可接受地保持准确性。此方法具体应用于微机械学，在微观结构中知道微观力学解，即给定异质微结构的变形和应力场，非常重要。我们的研究表明，基于物理的训练方法在未知微结构上相较于纯数据驱动方法具有更高的准确性。