AI 中文总结
针对计算力学中非线性参数化变分不等式，提出模型降阶框架，通过POD构建降阶模型，引入神经网络增强，用稀疏求积超降阶，在二维障碍和三维摩擦接触问题上验证，大幅降成本且逼近精确。
AI 中文摘要
我们为计算力学中出现的非线性参数化变分不等式提出了一个模型降阶框架。高维模型以混合原始对偶形式编写，并带有基于投影的互补条件，导致通过原始对偶形式的半光滑牛顿法求解非线性非光滑代数系统。在此基础上，通过对原始和对偶解快照进行适当正交分解（POD）构建降阶模型，并在降维空间中通过半光滑牛顿迭代求解所得的降阶系统。为了解决低维线性空间提供有限逼近效率的情况，我们引入了神经网络增强的降阶模型。两个前馈网络学习原始和对偶变量在截断POD坐标中的校正，定义了一个直接嵌入半光滑牛顿迭代中的非线性流形逼近。通过基于贪婪非负最小二乘法的稀疏求积方法进行超降阶，降低了与高维残差评估相关的在线成本。特别关注超降阶与学习到的非线性流形之间的相互作用。所提出的方法在两个具有不同非线性来源的非线性变分不等式上进行了评估：一个状态方程中具有三次非线性的二维障碍问题，以及一个库仑定律在约束方程中引起非线性投影的三维摩擦接触问题。数值结果比较了高维模型、线性降阶模型、神经网络增强降阶模型及其超降阶变体，证明了在大幅降低在线计算成本的情况下具有精确逼近。
英文摘要
We propose a model order reduction framework for nonlinear parametrized variational inequalities arising in computational mechanics. The high-dimensional model is written in mixed primal-dual form with projection-based complementarity conditions, leading to nonlinear nonsmooth algebraic systems solved by a semi-smooth Newton method in primal-dual form. On this basis, reduced models are constructed by proper orthogonal decomposition (POD) of both primal and dual solution snapshots, and the resulting reduced systems are solved by semi-smooth Newton iterations in the reduced space. To address cases where low-dimensional linear spaces provide limited approximation efficiency, we introduce a neural-network-augmented reduced model. Two feedforward networks learn corrections in the truncated POD coordinates of the primal and dual variables, defining a nonlinear manifold approximation that is embedded directly in the semi-smooth Newton iterations. The online cost associated with high-dimensional residual evaluations is reduced through hyper-reduction, using a sparse cubature approach based on greedy nonnegative least squares. Particular attention is paid to the interaction between hyper-reduction and the learned nonlinear manifold. The proposed methodology is assessed on two nonlinear variational inequalities with distinct sources of nonlinearity: a two-dimensional obstacle problem with a cubic nonlinearity in the state equation, and a three-dimensional frictional contact problem in which the Coulomb law induces a nonlinear projection in the constraint equation. Numerical results compare the high-dimensional model, the linear reduced model, the neural-network-augmented reduced model, and their hyper-reduced variants, demonstrating accurate approximations with substantial reductions in online computational cost.
