In this paper, we present a framework for learning the solution map of a backward parabolic Cauchy problem. The solution depends continuously but nonlinearly on the final data, source, and force terms, all residing in Banach spaces of functions. We utilize Fr\'echet space neural networks (Benth et al. (2023)) to address this operator learning problem. Our approach provides an alternative to Deep Operator Networks (DeepONets), using basis functions to span the relevant function spaces rather than relying on finite-dimensional approximations through censoring. With this method, structural information encoded in the basis coefficients is leveraged in the learning process. This results in a neural network designed to learn the mapping between infinite-dimensional function spaces. Our numerical proof-of-concept demonstrates the effectiveness of our method, highlighting some advantages over DeepONets.

翻译：本文提出了一种用于学习反向抛物型柯西问题解映射的框架。该解连续但非线性地依赖于终值数据、源项和力项，这些项均定义在函数的巴拿赫空间中。我们采用弗雷歇空间神经网络（Benth等人，2023）来处理这一算子学习问题。我们的方法为深度算子网络（DeepONets）提供了一种替代方案，通过基函数张成相关函数空间，而非依赖截断的有限维逼近。该方法使得基函数系数中编码的结构信息能够在学习过程中得到利用，从而构建出专门学习无限维函数空间之间映射的神经网络。数值概念验证展示了本方法的有效性，并凸显了其相较于DeepONets的若干优势。