Solving Parametric Partial Differential Equation Using Least-Squares-Based Neural Networks
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Developing efficient methods for solving parametric partial differential equations (PDEs) is crucial for addressing inverse problems. Herein, we propose a least-squares-based neural network method that utilizes a separated (low-rank) representation form of the parametric PDE solution. Specifically, we consider a vector-valued neural network, interpreted as a set of trainable basis functions, followed by a least-squares solver determining the discrete optimal coefficients of these basis functions for each given parameter. We call the resulting method the LS-Net method. We provide theoretical results similar to those of the Universal Approximation Theorem, stating that there exists a sufficiently large neural network that can approximate the parametric solution of a given parametric PDE up to a prescribed accuracy. During experimentation, we restrict to (Variational) Physics-Informed Neural Networks, although the presented framework applies to any PDE formulation with quadratic loss functions. Numerical results demonstrate the method's ability to approximate parametric solutions in one- and two-dimensional spatial problems.
