Deeper-PINNs: Unlocking the power of deep physics-informed neural networks

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Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a promising framework for solving partial differential equations (PDEs) and have garnered significant attention across industrial and scientific domains. However, their effectiveness is often constrained by limited approximation capacity and performance degradation in deep network structures. In this work, we propose the Deeper Physics-Informed Neural Network (Deeper-PINN), a novel architecture designed to address these challenges. The Deeper-PINN incorporates element-wise multiplication operations into the PINN structure, which effectively mitigates the initialization pathologies of PINNs and enables the utilization of deeper network structures. Additionally, this operation projects features into high-dimensional, nonlinear spaces, thereby enhancing the approximation capacity of PINNs. The proposed architecture is evaluated on multiple benchmark problems, demonstrating that Deeper-PINNs can effectively leverage deep neural network structures while maintaining high parameter efficiency. The complete codes of the experiments can be found on https://github.com/flongjiang/Deeper-PINNs • A novel architecture, Deeper-PINNs is developed that mitigates the degradation problem of deep PINNs. • Element-wise multiplication is introduced to mitigate the initialization pathology, enabling PINNs to effectively utilize deep neural network structures. • The developed Deeper-PINNs can map the features into nonlinear high-dimensional space, which enables Deeper-PINNs with better expressiveness.