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。

映射微结构的弹性特性到机械变形的有限算子学习技术

A finite operator learning technique for mapping the elastic properties  of microstructures to their mechanical deformations

We consider the parametric learning problem, where the objective of the
learner is determined by a parametric loss function. Employing empirical risk
minimization with possibly regularization, the inferred parameter vector will
be biased toward the training samples. Such bias is measured by the cross
validation procedure in practice where the data set is partitioned into a
training set used for training and a validation set, which is not used in
training and is left to measure the out-of-sample performance. A classical
cross validation strategy is the leave-one-out cross validation (LOOCV) where
one sample is left out for validation and training is done on the rest of the
samples that are presented to the learner, and this process is repeated on all
of the samples. LOOCV is rarely used in practice due to the high computational
complexity. In this paper, we first develop a computationally efficient
approximate LOOCV (ALOOCV) and provide theoretical guarantees for its
performance. Then we use ALOOCV to provide an optimization algorithm for
finding the regularizer in the empirical risk minimization framework. In our
numerical experiments, we illustrate the accuracy and efficiency of ALOOCV as
well as our proposed framework for the optimization of the regularizer.

本文提出了一种快速且准确的基于参数学习的交叉验证策略 ALOOCV，并利用其开发了一种基于经验风险最小化框架的正则化优化算法。