SIAM Journal on Imaging Sciences, Volume 14, Issue 1, Page 128-155, January 2021.
We introduce a novel neural network-based PDEs solver for forward and inverse problems. The solver is grid free, mesh free, and shape free, and the solution is approximated by a neural network. We employ an unsupervised approach such that the input to the network is a point set in an arbitrary domain, and the output is the set of the corresponding function values. The network is trained to minimize deviations of the learned function from the PDE solution and satisfy the boundary conditions. The resulting solution in turn is an explicit, smooth, differentiable function with a known analytical form. We solve the forward problem (observations given the underlying model's parameters), semi-inverse problem (model's parameters given the observations in the whole domain), and full tomography inverse problem (model's parameters given the observations on the boundary) by solving the forward and semi-inverse problems at the same time. The optimized loss function consists of few elements: fidelity term of $L_2$ norm that enforces the PDE in the weak sense, an $L_\infty$ norm term that enforces pointwise fidelity and thus promotes a strong solution, and boundary and initial conditions constraints. It further accommodates regularizers for the solution and/or the model's parameters of the differential operator. This setting is flexible in the sense that regularizers can be tailored to specific problems. We demonstrate our method on several free shape two dimensional (2D) second order systems with application to electrical impedance tomography (EIT) and diffusion equation. Unlike other numerical methods such as finite differences and finite elements, the derivatives of the desired function can be analytically calculated to any order. This framework enables, in principle, the solution of high order and high dimensional nonlinear PDEs.

中文翻译：

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基于无监督学习的基于PDE层析成像的强大解决方案

SIAM影像科学杂志，第14卷，第1期，第128-155页，2021年1月。
我们介绍了一种新颖的基于神经网络的PDE解算器，用于正向和逆向问题。求解器无网格，无网格和无形状，并且解决方案由神经网络近似。我们采用无监督的方法，以使网络的输入是在任意域中设置的点，而输出是相应功能值的集合。对网络进行训练，以使学习的功能与PDE解决方案的偏差最小，并满足边界条件。所得的解决方案又是具有已知分析形式的显式，平滑，可微函数。我们解决了前向问题（给定基础模型参数的观测值），半反问题（给定了整个域的观测值的模型参数）和全断层扫描反问题（模型' s参数给出边界上的观测值），同时解决正向和半反问题。优化的损失函数由以下几部分组成：$ L_2 $范数的保真度条件，用于在弱意义上实施PDE； $ L_ \ infty $范数，其强制实施逐点保真度，从而促进一个强解，以及边界条件和初始条件约束。它还为微分算子的解和/或模型参数提供了正则化器。在可以针对特定问题定制正则化程序的意义上，此设置很灵活。我们在几种自由形状的二维（2D）二阶系统上演示了我们的方法，并将其应用于电阻抗层析成像（EIT）和扩散方程。与其他数值方法（例如有限差分和有限元）不同，所需函数的导数可以解析为任何顺序。原则上，该框架可以实现高阶和高维非线性PDE的解决方案。