Neural Parameter Regression for Explicit Representations of PDE Solution Operators
This addresses the problem of efficiently solving PDEs for computational science and engineering, representing an incremental improvement over existing operator learning methods like DeepONets.
The paper tackles learning solution operators for Partial Differential Equations (PDEs) by introducing Neural Parameter Regression (NPR), which regresses neural network parameters using physics-informed techniques, resulting in enhanced parameter efficiency and adaptability to new conditions.
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.