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.

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，用于预测由偏微分方程（PDEs）控制的时空动态。该框架利用了受有限元方法（FEM）和隐式欧拉时间积分方案启发的损失函数。通过考虑瞬态导热传导问题进行性能测试，该算子学习框架将当前时间步的温度场作为输入，预测下一个时间步的温度场。实施物理学约束的热传导方程 Galerkin 离散弱形式被用作损失函数，称为有限算子学习（FOL）。经过训练，网络成功预测了任意初始温度场的时间演化，与有限元方法解决方案相比具有较高的准确性。该框架也适用于具有异质热导率和任意几何形状的情况。FOL 的优势可以概括为：首先，训练是以无监督的方式进行的，避免了需要准备大量昂贵模拟或实验数据集的需求。相反，使用由高斯随机过程和傅里叶级数生成的随机温度模式结合恒温场作为训练数据，以覆盖可能的温度情况。其次，利用形状函数和向后差分逼近进行域的离散化，得到一个纯代数方程。这提高了训练效率，同时避免了在优化权重和偏差时耗时的自动微分过程，虽然可能会导致离散化误差。最后，由于有限元方法的插值能力，FOL 能处理任意几何形状，这对于解决各种工程应用场景至关重要。