Revolutionizing High-Dimensional PDE Solutions: A Novel Approach of Integrated Truncated Singular Value Decomposition to Enhance Physics-Informed Neural Networks

Partial differential equations (PDEs) serve as crucial tools for modeling complex real-world phenomena. The physics-informed neural network (PINN) is an advanced method within artificial neural network (ANN)-based approaches for solving a wide range of PDE problems. The restarting strategy of PINN (r-PINN) is a novel modification that has demonstrated a significant reduction in computational time compared to its predecessor. However, the complexity of the r-PINN architecture poses challenges, particularly in accurately solving PDE problems with complex geometries, resulting in a large number of trainable parameters that affect computational performance and memory requirements. To address these challenges, this study integrates truncated singular value decomposition (TSVD) into both the PINN and r-PINN frameworks to uncover the underlying structure of the trainable parameter matrices, enabling their factorization into low-rank components. This process yields a compressed trainable parameter matrix, improving training efficiency while preserving accuracy. The proposed approach aims to obtain numerical solutions for various benchmark PDE problems, including up to three-dimensional cases and a PDE system. This approach is applicable not only to the novel r-PINN but also to the basic PINN, allowing for an analysis of the improvements gained through TSVD. Comparative analyses with non-TSVD approaches will be presented, and experimental results will demonstrate the efficacy of TSVD in enhancing computational performance across all benchmark problems, warranting further investigation.