Multiscale problems are ubiquitous in physics. Numerical simulations of such problems by solving partial differential equations (PDEs) at high resolution are computationally too expensive for many-query scenarios, such as uncertainty quantification, remeshing applications, and topology optimization. This limitation has motivated the development of data-driven surrogate models, where microscale computations are substituted by black-box mappings between macroscale quantities. While these approaches offer significant speedups, they typically struggle to incorporate microscale physical constraints, such as the balance of linear momentum. In this contribution, we propose the Equilibrium Neural Operator (EquiNO), a physics-informed PDE surrogate in which equilibrium is hard-enforced by construction. EquiNO achieves this by projecting the solution onto a set of divergence-free basis functions obtained via proper orthogonal decomposition (POD), thereby ensuring satisfaction of equilibrium without relying on penalty terms or multi-objective loss functions. We compare EquiNO with variational physics-informed neural and operator networks that enforce physical constraints only weakly through the loss function, as well as with purely data-driven operator-learning baselines. Our framework, applicable to multiscale FE$^{\,2}$ computations, introduces a finite element-operator learning (FE-OL) approach that integrates the finite element (FE) method with operator learning (OL). We apply the proposed methodology to quasi-static problems in solid mechanics and demonstrate that FE-OL yields accurate solutions even when trained on restricted datasets. The results show that EquiNO achieves speedup factors exceeding 8000-fold compared to traditional methods and offers a robust and physically consistent alternative to existing data-driven surrogate models.

翻译：多尺度问题在物理学中普遍存在。通过求解高分辨率偏微分方程（PDE）对此类问题进行数值模拟，在不确定性量化、重网格应用和拓扑优化等多查询场景中计算成本过高。这一局限性推动了数据驱动代理模型的发展，其中微观尺度计算被宏观量之间的黑箱映射所替代。虽然这些方法能显著加速计算，但它们通常难以纳入微观尺度的物理约束，例如线性动量平衡。在本文中，我们提出了平衡神经算子（EquiNO），一种通过构造硬性强制满足平衡条件的物理信息PDE代理模型。EquiNO通过将解投影到一组通过本征正交分解（POD）获得的无散基函数上来实现这一点，从而确保满足平衡条件，而无需依赖惩罚项或多目标损失函数。我们将EquiNO与仅通过损失函数弱式强制物理约束的变分物理信息神经网络及算子网络，以及纯数据驱动的算子学习基线方法进行了比较。我们的框架适用于多尺度FE$^{\,2}$计算，引入了一种将有限元（FE）方法与算子学习（OL）相结合的有限元-算子学习（FE-OL）方法。我们将所提出的方法应用于固体力学中的准静态问题，并证明即使基于有限数据集进行训练，FE-OL也能产生精确解。结果表明，与传统方法相比，EquiNO实现了超过8000倍的加速，并为现有数据驱动代理模型提供了一个鲁棒且物理一致的替代方案。