The solution of probabilistic inverse problems for which the corresponding forward problem is constrained by physical principles is challenging. This is especially true if the dimension of the inferred vector is large and the prior information about it is in the form of a collection of samples. In this work, a novel deep learning based approach is developed and applied to solving these types of problems. The approach utilizes samples of the inferred vector drawn from the prior distribution and a physics-based forward model to generate training data for a conditional Wasserstein generative adversarial network (cWGAN). The cWGAN learns the probability distribution for the inferred vector conditioned on the measurement and produces samples from this distribution. The cWGAN developed in this work differs from earlier versions in that its critic is required to be 1-Lipschitz with respect to both the inferred and the measurement vectors and not just the former. This leads to a loss term with the full (and not partial) gradient penalty. It is shown that this rather simple change leads to a stronger notion of convergence for the conditional density learned by the cWGAN and a more robust and accurate sampling strategy. Through numerical examples it is shown that this change also translates to better accuracy when solving inverse problems. The numerical examples considered include illustrative problems where the true distribution and/or statistics are known, and a more complex inverse problem motivated by applications in biomechanics.

翻译：概率反问题的求解面临挑战，因为其对应的正问题受物理原理约束，尤其是在推断向量维度较大且先验信息以样本集合形式给出时。本文提出并应用了一种基于深度学习的新方法来解决这类问题。该方法利用从先验分布中抽取的推断向量样本及基于物理的正向模型，为条件Wasserstein生成对抗网络（cWGAN）生成训练数据。该cWGAN学习给定测量条件下推断向量的概率分布，并生成该分布的样本。本文开发的cWGAN与早期版本的不同之处在于，其判别器需同时针对推断向量和测量向量满足1-Lipschitz条件（而非仅针对前者），从而引入包含全梯度惩罚（而非部分梯度惩罚）的损失项。研究表明，这一简单修改使cWGAN学习的条件密度具有更强的收敛性，并带来更稳健、更精确的采样策略。通过数值示例证明，该修改在求解反问题时也能转化为更高的精度。所考虑的数值示例包括已知真实分布和/或统计量的说明性问题，以及受生物力学应用启发的更复杂反问题。