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.

翻译：算子学习框架的预测精度依赖于可用训练数据（输入-输出函数对）的质量与数量，通常需要大量高保真度数据，而这在某些实际工程应用中难以获取。这些数据集在不同实现间可能具有非均匀离散化特征，网格分辨率随样本变化。本研究通过将分辨率无关神经算子（RINO）框架扩展至完全无数据设置，提出一种物理信息驱动的算子学习方法，以同时应对上述挑战。该方法将任意（但足够精细）离散化的输入函数通过预训练的基函数投影至潜在嵌入空间（即有限维向量空间）。随后，与底层偏微分方程（PDE）相关的算子通过一个简单的多层感知机（MLP）进行近似，该感知机以潜在编码及时空坐标作为输入，在物理空间中生成解。PDE约束通过物理空间中的有限差分求解器实现。所提方法的验证与性能在多个具有多分辨率数据的数值算例中进行基准测试，其中输入函数以包括粗粒度与细粒度离散化在内的不同分辨率进行采样。