This article introduces a novel approach to learning monotone neural networks
through a newly defined penalization loss. The proposed method is particularly
effective in solving classes of variational problems, specifically monotone
inclusion problems, commonly encountered in image processing tasks. The
Forward-Backward-Forward (FBF) algorithm is employed to address these problems,
offering a solution even when the Lipschitz constant of the neural network is
unknown. Notably, the FBF algorithm provides convergence guarantees under the
condition that the learned operator is monotone. Building on plug-and-play
methodologies, our objective is to apply these newly learned operators to
solving non-linear inverse problems. To achieve this, we initially formulate
the problem as a variational inclusion problem. Subsequently, we train a
monotone neural network to approximate an operator that may not inherently be
monotone. Leveraging the FBF algorithm, we then show simulation examples where
the non-linear inverse problem is successfully solved.

提出了一种新的学习单调神经网络的方法，通过一种新定义的惩罚损失函数。该方法在解决一类变分问题中表现出特别有效性，尤其是在图像处理任务中经常遇到的单调包含问题。使用前向 - 后向 - 前向（FBF）算法来解决这些问题，即使神经网络的利普希茨常数未知，也能提供解决方案。值得注意的是，FBF 算法在学习的算子是单调的条件下提供收敛保证。基于即插即用方法，我们的目标是将这些新学习的算子应用于解决非线性反问题。为此，我们首先将问题制定为一个变分包含问题。随后，我们训练一个单调神经网络来逼近一个可能不是单调的算子。利用 FBF 算法，我们展示了成功解决非线性反问题的仿真例子。