The Deep Operator Network (DeepONet) is a powerful neural operator architecture that uses two neural networks to map between infinite-dimensional function spaces. This architecture allows for the evaluation of the solution field at any location within the domain but requires input functions to be discretized at identical locations, limiting practical applications. We introduce a general framework for operator learning from input-output data with arbitrary sensor locations and counts. This begins by introducing a resolution-independent DeepONet (RI-DeepONet), which handles input functions discretized arbitrarily but sufficiently finely. To achieve this, we propose two dictionary learning algorithms that adaptively learn continuous basis functions, parameterized as implicit neural representations (INRs), from correlated signals on arbitrary point clouds. These basis functions project input function data onto a finite-dimensional embedding space, making it compatible with DeepONet without architectural changes. We specifically use sinusoidal representation networks (SIRENs) as trainable INR basis functions. Similarly, the dictionary learning algorithms identify basis functions for output data, defining a new neural operator architecture: the Resolution Independent Neural Operator (RINO). In RINO, the operator learning task reduces to mapping coefficients of input basis functions to output basis functions. We demonstrate RINO's robustness and applicability in handling arbitrarily sampled input and output functions during both training and inference through several numerical examples.

翻译：深度算子网络（DeepONet）是一种强大的神经算子架构，它使用两个神经网络在无限维函数空间之间进行映射。该架构允许在域内任意位置评估解场，但要求输入函数在相同位置进行离散化，这限制了实际应用。我们提出了一个从具有任意传感器位置和数量的输入-输出数据中学习算子的通用框架。首先引入分辨率无关的DeepONet（RI-DeepONet），它可以处理任意但足够精细离散化的输入函数。为实现这一点，我们提出了两种字典学习算法，它们能够从任意点云上的相关信号中自适应地学习连续基函数，这些基函数被参数化为隐式神经表示（INR）。这些基函数将输入函数数据投影到有限维嵌入空间，使其无需架构修改即可与DeepONet兼容。我们特别使用正弦表示网络（SIREN）作为可训练的INR基函数。类似地，字典学习算法为输出数据识别基函数，从而定义了一种新的神经算子架构：分辨率无关神经算子（RINO）。在RINO中，算子学习任务简化为将输入基函数的系数映射到输出基函数的系数。通过多个数值算例，我们证明了RINO在训练和推理过程中处理任意采样的输入和输出函数时具有鲁棒性和适用性。