We propose a novel composite framework to find unknown fields in the context of inverse problems for partial differential equations (PDEs). We blend the high expressibility of deep neural networks as universal function estimators with the accuracy and reliability of existing numerical algorithms for partial differential equations as custom layers in semantic autoencoders. Our design brings together techniques of computational mathematics, machine learning and pattern recognition under one umbrella to incorporate domain-specific knowledge and physical constraints to discover the underlying hidden fields. The network is explicitly aware of the governing physics through a hard-coded PDE solver layer in contrast to most existing methods that incorporate the governing equations in the loss function or rely on trainable convolutional layers to discover proper discretizations from data. This subsequently focuses the computational load to only the discovery of the hidden fields and therefore is more data efficient. We call this architecture Blended inverse-PDE networks (hereby dubbed BiPDE networks) and demonstrate its applicability for recovering the variable diffusion coefficient in Poisson problems in one and two spatial dimensions, as well as the diffusion coefficient in the time-dependent and nonlinear Burgers' equation in one dimension. We also show that the learned hidden parameters are robust to added noise on input data.

我们提出了一种新颖的复合框架，可以在偏微分方程（PDE）的逆问题的上下文中找到未知字段。我们将作为通用函数估计器的深层神经网络的高表达能力与作为语义自动编码器中自定义层的偏微分方程的现有数值算法的准确性和可靠性相结合。我们的设计将计算数学，机器学习和模式识别等技术融为一体，以融合特定领域的知识和物理约束来发现潜在的隐藏领域。与大多数在损耗函数中包含控制方程或依靠可训练卷积层从数据中发现适当离散的现有方法相比，该网络通过硬编码的PDE求解器层明确了解控制物理。随后，这会将计算负荷集中在仅隐藏字段的发现上，因此数据效率更高。我们将这种架构称为混合逆PDE网络（以下称为BiPDE网络），并证明其可用于恢复一维和二维空间中泊松问题中的可变扩散系数，以及时变和非线性Burgers模型中的扩散系数。一维方程。我们还表明，学习到的隐藏参数对于增加输入数据的噪声具有鲁棒性。