We propose a novel finite element-based physics-informed operator learning framework that allows for predicting spatiotemporal dynamics governed by partial differential equations (PDEs). The proposed framework employs a loss function inspired by the finite element method (FEM) with the implicit Euler time integration scheme. A transient thermal conduction problem is considered to benchmark the performance. The proposed operator learning framework takes a temperature field at the current time step as input and predicts a temperature field at the next time step. The Galerkin discretized weak formulation of the heat equation is employed to incorporate physics into the loss function, which is coined finite operator learning (FOL). Upon training, the networks successfully predict the temperature evolution over time for any initial temperature field at high accuracy compared to the FEM solution. The framework is also confirmed to be applicable to a heterogeneous thermal conductivity and arbitrary geometry. The advantages of FOL can be summarized as follows: First, the training is performed in an unsupervised manner, avoiding the need for a large data set prepared from costly simulations or experiments. Instead, random temperature patterns generated by the Gaussian random process and the Fourier series, combined with constant temperature fields, are used as training data to cover possible temperature cases. Second, shape functions and backward difference approximation are exploited for the domain discretization, resulting in a purely algebraic equation. This enhances training efficiency, as one avoids time-consuming automatic differentiation when optimizing weights and biases while accepting possible discretization errors. Finally, thanks to the interpolation power of FEM, any arbitrary geometry can be handled with FOL, which is crucial to addressing various engineering application scenarios.

翻译：我们提出了一种新颖的基于有限元的物理信息算子学习框架，可用于预测由偏微分方程（PDEs）支配的时空动力学。该框架采用受有限元法（FEM）及隐式欧拉时间积分格式启发的损失函数。本文以瞬态热传导问题作为性能基准。所提出的算子学习框架以当前时间步的温度场作为输入，并预测下一时间步的温度场。通过采用热方程的Galerkin离散弱形式将物理规律融入损失函数，该方法被称为有限算子学习（FOL）。训练完成后，网络能够针对任意初始温度场，以相较于FEM解的高精度成功预测温度随时间演化。该框架亦被证实适用于非均匀热导率及任意几何形状。FOL的优势可总结如下：首先，训练以无监督方式进行，避免了依赖从昂贵仿真或实验中获取的大规模数据集。取而代之的是，采用高斯随机过程与傅里叶级数生成的随机温度场，结合恒定温度场作为训练数据，以覆盖可能的温度分布情况。其次，利用形函数与后向差分近似进行区域离散化，得到纯代数方程。这提升了训练效率，因为在优化权重与偏置时避免了耗时的自动微分，同时接受了可能的离散误差。最后，得益于FEM的插值能力，FOL可处理任意几何形状，这对应对多样化的工程应用场景至关重要。