We propose a new neural network based method for solving inverse problems for partial differential equations (PDEs) by formulating the PDE inverse problem as a bilevel optimization problem. At the upper level, we minimize the data loss with respect to the PDE parameters. At the lower level, we train a neural network to locally approximate the PDE solution operator in the neighborhood of a given set of PDE parameters, which enables an accurate approximation of the descent direction for the upper level optimization problem. The lower level loss function includes the L2 norms of both the residual and its derivative with respect to the PDE parameters. We apply gradient descent simultaneously on both the upper and lower level optimization problems, leading to an effective and fast algorithm. The method, which we refer to as BiLO (Bilevel Local Operator learning), is also able to efficiently infer unknown functions in the PDEs through the introduction of an auxiliary variable. We demonstrate that our method enforces strong PDE constraints, is robust to sparse and noisy data, and eliminates the need to balance the residual and the data loss, which is inherent to soft PDE constraints.

翻译：我们提出了一种基于神经网络的新方法，通过将偏微分方程（PDE）反问题表述为双层优化问题来求解。在上层，我们最小化关于PDE参数的数据损失。在下层，我们训练一个神经网络，在给定PDE参数集邻域内局部逼近PDE解算子，从而能够精确逼近上层优化问题的下降方向。下层损失函数包括残差及其关于PDE参数导数的L2范数。我们同时对上层和下层优化问题应用梯度下降，从而形成一种高效快速的算法。该方法被称为BiLO（双层局部算子学习），还能通过引入辅助变量有效推断PDE中的未知函数。我们证明，该方法施加了强PDE约束，对稀疏和含噪声数据具有鲁棒性，并且消除了软PDE约束中固有的残差与数据损失之间的平衡需求。