This work presents a finite element-guided physics-informed operator learning framework for multiphysics problems with coupled partial differential equations (PDEs) on arbitrary domains. The proposed framework learns an operator from the input space to the solution space with a weighted residual formulation based on the finite element method, enabling discretization-independent prediction beyond the training resolution without relying on labeled simulation data. The present framework for multiphysics problems is implemented in Folax, a JAX-based operator learning platform, and is verified on nonlinear coupled thermo-mechanical problems. Two- and three-dimensional representative volume elements with varying heterogeneous microstructures, and a close-to-reality industrial casting example under varying boundary conditions are investigated as the example problems. We investigate the potential of several neural operators combined with the proposed finite element-guided approach, including Fourier neural operators (FNOs), deep operator networks (DeepONets), and a newly proposed implicit finite operator learning (iFOL) approach based on conditional neural fields. The results demonstrate that FNOs yield highly accurate solution operators on regular domains, where the global features can be efficiently learned in the spectral domain, and iFOL offers efficient parametric operator learning capabilities for complex and irregular geometries. Furthermore, studies on training strategies, network decomposition, and training sample quality reveal that a monolithic training strategy using a single network is sufficient for accurate predictions, while training sample quality strongly influences performance. Overall, the present approach highlights the potential of physics-informed operator learning with a finite element-based loss as a unified and scalable approach for coupled multiphysics simulations.

翻译：本文提出了一种基于有限元引导的物理信息算子学习框架，用于求解任意域上具有耦合偏微分方程（PDEs）的多物理场问题。该框架利用基于有限元方法的加权残差公式，从输入空间到解空间学习算子，能够在无需依赖标注仿真数据的情况下，实现超越训练分辨率的离散无关预测。该多物理场问题求解框架在基于JAX的算子学习平台Folax中实现，并在非线性耦合热-力学问题上得到验证。以二维和三维具有不同异质微结构的代表性体积单元，以及变边界条件下接近真实工业场景的铸造案例作为示例问题。我们研究了多种神经算子与所提有限元引导方法结合的潜力，包括傅里叶神经算子（FNOs）、深度算子网络（DeepONets）以及一种基于条件神经场的新提出的隐式有限元算子学习（iFOL）方法。结果表明，FNOs在规则域上能够生成高精度解算子，其全局特征可在谱域中高效学习；而iFOL则为复杂和不规则几何提供了高效的参数化算子学习能力。此外，关于训练策略、网络分解和训练样本质量的研究表明，采用单一网络的整体训练策略足以实现精确预测，而训练样本质量对性能影响显著。总体而言，本方法凸显了基于有限元损失的物理信息算子学习作为耦合多物理场仿真的一种统一且可扩展的方法的潜力。