Thanks to their universal approximation properties and new efficient training strategies, Deep Neural Networks are becoming a valuable tool for the approximation of mathematical operators. In the present work, we introduce Mesh-Informed Neural Networks (MINNs), a class of architectures specifically tailored to handle mesh based functional data, and thus of particular interest for reduced order modeling of parametrized Partial Differential Equations (PDEs). The driving idea behind MINNs is to embed hidden layers into discrete functional spaces of increasing complexity, obtained through a sequence of meshes defined over the underlying spatial domain. The approach leads to a natural pruning strategy which enables the design of sparse architectures that are able to learn general nonlinear operators. We assess this strategy through an extensive set of numerical experiments, ranging from nonlocal operators to nonlinear diffusion PDEs, where MINNs are compared against more traditional architectures, such as classical fully connected Deep Neural Networks, but also more recent ones, such as DeepONets and Fourier Neural Operators. Our results show that MINNs can handle functional data defined on general domains of any shape, while ensuring reduced training times, lower computational costs, and better generalization capabilities, thus making MINNs very well-suited for demanding applications such as Reduced Order Modeling and Uncertainty Quantification for PDEs.

翻译：得益于其通用逼近特性与新型高效训练策略，深度神经网络已成为数学算子近似的重要工具。本文提出网格信息神经网络（MINNs），这是一类专为处理基于网格的函数数据而设计的架构，对于参数化偏微分方程的降阶建模具有特殊意义。MINNs的核心思想是将隐藏层嵌入到通过底层空间域上定义的网格序列获得的、复杂度递增的离散函数空间中。该方法导出的自然剪枝策略能够设计出可学习通用非线性算子的稀疏架构。我们通过涵盖非局部算子到非线性扩散偏微分方程的大量数值实验评估该策略，并将MINNs与传统架构（如经典全连接深度神经网络）及近年新架构（如深度算子网络和傅里叶神经算子）进行对比。结果表明，MINNs可处理定义在任意形状一般域上的函数数据，同时实现更短的训练时间、更低的计算成本和更优的泛化能力，使其非常适用于偏微分方程降阶建模与不确定性量化等强需求应用场景。