标题： 基于物理信息的多分辨率神经算子 显示英文标题

标题： A Physics-informed Multi-resolution Neural Operator

摘要： 操作学习框架的预测准确性取决于可用训练数据（输入-输出函数对）的质量和数量，通常需要大量高保真数据，这在一些实际工程应用中可能难以获得。 这些数据集可能在不同实现之间离散化不均匀，网格分辨率在样本之间变化。 在本研究中，我们通过将分辨率无关神经算子（RINO）框架扩展到完全数据自由的设置，引入了一种物理信息操作学习方法，同时解决了这两个挑战。 在这里，任意（但足够精细）离散化的输入函数使用预训练的基函数投影到潜在嵌入空间（即有限维向量空间）。 然后，与底层偏微分方程（PDEs）相关的算子由一个简单的多层感知机（MLP）近似，该感知机以潜在代码和时空坐标作为输入，在物理空间中生成解。 通过在物理空间中的有限差分求解器强制执行PDEs。 所提出的方法在多个具有多分辨率数据的数值示例上进行了验证和性能测试，其中输入函数在不同分辨率下进行采样，包括粗略和精细的离散化。 显示英文摘要

摘要： The predictive accuracy of operator learning frameworks depends on the quality and quantity of available training data (input-output function pairs), often requiring substantial amounts of high-fidelity data, which can be challenging to obtain in some real-world engineering applications. These datasets may be unevenly discretized from one realization to another, with the grid resolution varying across samples. In this study, we introduce a physics-informed operator learning approach by extending the Resolution Independent Neural Operator (RINO) framework to a fully data-free setup, addressing both challenges simultaneously. Here, the arbitrarily (but sufficiently finely) discretized input functions are projected onto a latent embedding space (i.e., a vector space of finite dimensions), using pre-trained basis functions. The operator associated with the underlying partial differential equations (PDEs) is then approximated by a simple multi-layer perceptron (MLP), which takes as input a latent code along with spatiotemporal coordinates to produce the solution in the physical space. The PDEs are enforced via a finite difference solver in the physical space. The validation and performance of the proposed method are benchmarked on several numerical examples with multi-resolution data, where input functions are sampled at varying resolutions, including both coarse and fine discretizations.