发表机构
Northwestern University; HIDENN-AI, INC; University of Maryland, Baltimore County; University of Texas at Austin（西北大学; HIDENN-AI 公司; 马里兰大学巴尔的摩县分校; 德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出插值神经算子（INO），一种无数据、基于弱形式训练的神经网络，通过贪婪交替最小二乘实现高效训练，在多个基准上以更高精度和更快速度超越物理信息FNO和DeepONet。
AI 中文摘要
神经算子已成为逼近参数化偏微分方程（PDE）解算子的一种流行方法。然而，现有的神经算子要么需要大量的模拟数据，要么需要在GPU上进行长时间的物理信息训练，并且它们无法告知单个预测的准确程度。在本文中，我们提出了插值神经算子（INO），一种无数据的插值神经网络，它直接基于PDE的弱形式进行训练。在INO中，输入场的Karhunen-Loève坐标与空间坐标一起作为额外输入处理，每个输入由一个C-HiDeNN子网络逼近，其可训练参数为节点值。由于网络在其参数上是多线性的，训练通过贪婪交替最小二乘法简化为一系列一维线性求解。因此，INO在单个CPU核心上训练，并在微秒内预测新解。对于强制性问题，每个预测的总误差由一个可计算的残差界限所界定，该界限不需要参考解，并且相同的界限适用于其他精确满足边界条件的方法的预测。在每次预测之前，INO检查输入的领先坐标是否位于其训练范围内，范围之外的输入可以传递给传统求解器或传递给在更宽范围内训练的INO。INO与物理信息FNO和DeepONet在不同基准上进行了比较。INO在大多数这些问题上是最准确的模型，在二维Helmholtz问题（65^2网格）上准确度提高15倍，在扩散-反应基准上提高53倍，并且在一维和二维问题上，其在单个CPU核心上的训练时间比物理信息基线在单个GPU上的训练时间少3-80倍。
英文摘要
Neural operators have become a popular approach to approximate the solution operators of parametric partial differential equations (PDEs). However, existing neural operators either require a large amount of simulation data or a long physics-informed training on GPUs, and they cannot tell how accurate an individual prediction is. In this paper, we propose the Interpolating Neural Operator (INO), a data-free interpolating neural network that is trained directly on the weak form of the PDE. In INO, the Karhunen-Loève coordinates of the input field are treated as additional inputs together with the spatial coordinates, and each input is approximated by a C-HiDeNN sub-network whose trainable parameters are nodal values. Since the network is multilinear in its parameters, training reduces to a sequence of one-dimensional linear solves by greedy alternating least squares. As a result, INO trains on one CPU core and predicts a new solution in microseconds. For coercive problems, the total error of every prediction is bounded by a computable residual bound that requires no reference solution, and the same bound applies to the predictions of other methods that satisfy the boundary conditions exactly. Before each prediction, INO checks whether the leading coordinates of the input lie within the range on which it is trained, and inputs outside this range can be passed to a conventional solver or to an INO trained on a wider range. INO is compared with physics-informed FNO and DeepONet on different benchmarks. INO is the most accurate model on most of these problems, by 15$\times$ on two-dimensional Helmholtz at $65^2$ and 53$\times$ on the diffusion-reaction benchmark, and on the one- and two-dimensional problems its training on one CPU core takes 3-80$\times$ less time than the physics-informed baselines on one GPU.
