标题： 通过深度神经网络方法求解抛物方程中的反演势问题 显示英文标题

标题： Solving the inverse potential problem in the parabolic equation by the deep neural networks method

摘要： 在本工作中，我们考虑抛物方程中的一个反演势问题，其中未知势是一个与空间相关的函数，使用的测量数据是最终时间的数据。 该反演问题中的未知势通过深度神经网络（DNNs）进行参数化，以实现重建方案。 首先，在对输入源的一些正则性假设下证明了反演问题的唯一性。 然后，我们提出了一种新的损失函数，该函数包含依赖于偏微分方程（PDEs）残差和测量数据导数的正则化项。 这些额外项有效地在解中引入了更高的正则性，从而可以处理反演问题的不适定性。 此外，我们严格建立了相应的泛化误差估计。 我们的证明利用了经典线性反演源问题的条件稳定性，以及对噪声测量数据的平滑处理，以减少扰动误差。 最后，提供了数值算法和一些数值结果。 显示英文摘要

摘要： In this work, we consider an inverse potential problem in the parabolic equation, where the unknown potential is a space-dependent function and the used measurement is the final time data. The unknown potential in this inverse problem is parameterized by deep neural networks (DNNs) for the reconstruction scheme. First, the uniqueness of the inverse problem is proved under some regularities assumption on the input sources. Then we propose a new loss function with regularization terms depending on the derivatives of the residuals for partial differential equations (PDEs) and the measurements. These extra terms effectively induce higher regularity in solutions so that the ill-posedness of the inverse problem can be handled. Moreover, we establish the corresponding generalization error estimates rigorously. Our proofs exploit the conditional stability of the classical linear inverse source problems, and the mollification on the noisy measurement data which is set to reduce the perturbation errors. Finally, the numerical algorithm and some numerical results are provided.