Operator regression provides a powerful means of constructing discretization-invariant emulators for partial-differential equations (PDEs) describing physical systems. Neural operators specifically employ deep neural networks to approximate mappings between infinite-dimensional Banach spaces. As data-driven models, neural operators require the generation of labeled observations, which in cases of complex high-fidelity models result in high-dimensional datasets containing redundant and noisy features, which can hinder gradient-based optimization. Mapping these high-dimensional datasets to a low-dimensional latent space of salient features can make it easier to work with the data and also enhance learning. In this work, we investigate the latent deep operator network (L-DeepONet), an extension of standard DeepONet, which leverages latent representations of high-dimensional PDE input and output functions identified with suitable autoencoders. We illustrate that L-DeepONet outperforms the standard approach in terms of both accuracy and computational efficiency across diverse time-dependent PDEs, e.g., modeling the growth of fracture in brittle materials, convective fluid flows, and large-scale atmospheric flows exhibiting multiscale dynamical features.

翻译：算子回归为描述物理系统的偏微分方程构建离散不变仿真器提供了强大手段。神经算子通过深度神经网络逼近无限维巴拿赫空间之间的映射。作为数据驱动模型，神经算子需要生成带标签的观测数据，而复杂高保真模型产生的数据集往往包含冗余噪声特征，可能阻碍梯度优化。将高维数据集映射到低维潜在空间的有效特征上，既能简化数据处理又能增强学习效果。本研究提出潜在深度算子网络（L-DeepONet），该模型扩展了标准DeepONet，利用适合的自编码器识别高维偏微分方程输入/输出函数的潜在表示。我们证明，在模拟脆性材料断裂扩展、对流流体流动及展现多尺度动力学特征的大尺度大气流动等多类时变偏微分方程中，L-DeepONet在精度和计算效率上均优于标准方法。