We introduce Neural Parameter Regression (NPR), a novel framework specifically developed for learning solution operators in Partial Differential Equations (PDEs). Tailored for operator learning, this approach surpasses traditional DeepONets (Lu et al., 2021) by employing Physics-Informed Neural Network (PINN, Raissi et al., 2019) techniques to regress Neural Network (NN) parameters. By parametrizing each solution based on specific initial conditions, it effectively approximates a mapping between function spaces. Our method enhances parameter efficiency by incorporating low-rank matrices, thereby boosting computational efficiency and scalability. The framework shows remarkable adaptability to new initial and boundary conditions, allowing for rapid fine-tuning and inference, even in cases of out-of-distribution examples.

翻译：我们提出一种名为神经参数回归（NPR）的新型框架，专门用于学习偏微分方程（PDE）中的解算子。该方法针对算子学习量身定制，通过采用物理信息神经网络（PINN，Raissi等人，2021）技术回归神经网络（NN）参数，超越了传统的DeepONets（Lu等人，2021）。通过根据特定初始条件参数化每个解，该方法有效近似了函数空间之间的映射。我们的方法通过引入低秩矩阵提升参数效率，从而增强计算效率与可扩展性。该框架对新的初始条件和边界条件展现出卓越的适应性，支持快速微调与推理，即便在面对分布外样本时亦能应对。