ADCME is a novel computational framework to solve inverse problems involving physical simulations and deep neural networks (DNNs). This paper benchmarks its capability to learn spatially-varying physical fields using DNNs. We demonstrate that our approach has superior accuracy compared to the discretization approach on a variety of problems, linear or nonlinear, static or dynamic. Technically, we formulate our inverse problem as a PDE-constrained optimization problem. We express both the numerical simulations and DNNs using computational graphs and therefore, we can calculate the gradients using reverse-mode automatic differentiation. We apply a physics constrained learning algorithm (PCL) to efficiently back-propagate gradients through iterative solvers for nonlinear equations. The open-source software which accompanies the present paper can be found at https://github.com/kailaix/ADCME.jl.

中文翻译：

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ADCME：使用深度神经网络学习空间变化的物理场

ADCME是解决涉及物理模拟和深度神经网络（DNN）的逆问题的新颖计算框架。本文对使用DNN学习空间变化的物理场的能力进行了基准测试。我们证明，与离散化方法相比，在线性或非线性，静态或动态问题上，我们的方法具有更高的准确性。从技术上讲，我们将逆问题公式化为PDE约束的优化问题。我们使用计算图表示数值模拟和DNN，因此，我们可以使用反向模式自动微分来计算梯度。我们将物理约束学习算法（PCL）应用于通过非线性方程的迭代求解器有效地反向传播梯度。