Neural Parameter Regression for Explicit Representations of PDE Solution Operators

Konrad Mundinger, Max Zimmer, Sebastian Pokutta

Keywords: Physics-Informed Neural Networks, Partial Differential Equations, Operator Learning, Hypernetworks, Scientific Machine Learning

TL;DR: This paper presents Neural Parameter Regression, a novel approach that leverages Hypernetworks and low-rank matrix approximations to enhance Physics-Informed Neural Networks (PINNs) for efficient and adaptable learning of PDE solution operators.

Abstract: 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., 2021a) 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.

Submission Number: 10