We present a method that employs physics-informed deep learning techniques
for parametrically solving partial differential equations. The focus is on the
steady-state heat equations within heterogeneous solids exhibiting significant
phase contrast. Similar equations manifest in diverse applications like
chemical diffusion, electrostatics, and Darcy flow. The neural network aims to
establish the link between the complex thermal conductivity profiles and
temperature distributions, as well as heat flux components within the
microstructure, under fixed boundary conditions. A distinctive aspect is our
independence from classical solvers like finite element methods for data. A
noteworthy contribution lies in our novel approach to defining the loss
function, based on the discretized weak form of the governing equation. This
not only reduces the required order of derivatives but also eliminates the need
for automatic differentiation in the construction of loss terms, accepting
potential numerical errors from the chosen discretization method. As a result,
the loss function in this work is an algebraic equation that significantly
enhances training efficiency. We benchmark our methodology against the standard
finite element method, demonstrating accurate yet faster predictions using the
trained neural network for temperature and flux profiles. We also show higher
accuracy by using the proposed method compared to purely data-driven approaches
for unforeseen scenarios.

利用物理知识驱动的深度学习方法在异质固体中解决参数化偏微分方程，它的关键是建立复杂的热导率分布、温度分布和热流分量之间的联系，通过固定边界条件，在这项工作中，我们独立于有限元方法等经典求解器，并通过基于离散弱形式的损失函数定义方法给出出色的结果，该损失函数是一个代数方程，大大提高了训练效率。通过将我们的方法与标准有限元方法进行基准测试，我们展示了使用训练有素的神经网络在温度和通量剖面方面进行准确且更快的预测，我们还展示了在未知情况下，与纯数据驱动方法相比，所提出的方法具有更高的准确性。