Neural Parameter Regression for Explicit Representations of PDE Solution
  Operators
Konrad Mundinger Max Zimmer Sebastian Pokutta
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.