论文标题
部分可观测时空混沌系统的无模型预测
Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces
论文作者
论文摘要
在这项工作中，我们介绍和研究连续的生成神经网络（CGNN），即连续设置中的生成模型：CGNN的输出属于无限二维功能空间。该体系结构的灵感来自DCGAN，具有一个完全连接的层，几个卷积层和非线性激活函数。在连续的$ l^2 $设置中，每一层空间的尺寸被紧凑型小波的多分辨率分析的尺度所取代。我们介绍了卷积过滤器的条件以及保证CGNN具有侵入性的非线性。该理论发现了对反问题的应用，并允许得出（可能是非线性的）无限维逆问题的Lipschitz稳定性估计，其属于CGNN产生的流形的未知数。几个数值模拟，包括信号脱毛，说明和验证这种方法。
In this work, we present and study Continuous Generative Neural Networks (CGNNs), namely, generative models in the continuous setting: the output of a CGNN belongs to an infinite-dimensional function space. The architecture is inspired by DCGAN, with one fully connected layer, several convolutional layers and nonlinear activation functions. In the continuous $L^2$ setting, the dimensions of the spaces of each layer are replaced by the scales of a multiresolution analysis of a compactly supported wavelet. We present conditions on the convolutional filters and on the nonlinearity that guarantee that a CGNN is injective. This theory finds applications to inverse problems, and allows for deriving Lipschitz stability estimates for (possibly nonlinear) infinite-dimensional inverse problems with unknowns belonging to the manifold generated by a CGNN. Several numerical simulations, including signal deblurring, illustrate and validate this approach.
