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.

为了学习偏微分方程（PDEs）中的解算符，我们引入了神经参数回归（NPR）这一新颖框架。通过采用物理知识引导神经网络（PINN）技术对神经网络（NN）参数进行回归，该方法超越了传统的 DeepONets。我们的方法通过在每个解中基于特定初始条件进行参数化，有效地近似了函数空间之间的映射。通过引入低秩矩阵，我们的方法提高了参数效率，从而提升了计算效率和可扩展性。该框架在面对新的初始和边界条件时显示出显著的适应性，即使在分布之外的示例情况下也能进行快速微调和推断。