Least-Squares-Embedded Optimization for Accelerated Convergence of PINNs in Acoustic Wavefield Simulations

2025-04-23

AI Summary

To address the slow convergence and instability of gradient descent (GD) in physics-informed neural networks (PINNs) for solving the Helmholtz equation in the frequency domain—particularly at high frequencies—this paper proposes a hybrid optimization framework integrating GD with least-squares (LS) minimization. Specifically, a lightweight LS solver is embedded directly into the PINN loss function, performing closed-form optimal updates solely at the output layer. This constitutes the first end-to-end coupling of LS optimization with the PINN loss, supporting both scenarios with and without perfectly matched layers (PML). By operating on a compact normal matrix, the LS component incurs low computational overhead and scales efficiently. Numerical experiments on benchmark velocity models demonstrate significantly accelerated convergence, improved accuracy, and enhanced stability over standard GD—especially in high-frequency scattering simulations, where robustness is consistently maintained.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Optimization
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Physics-Informed Neural Networks (PINNs) have shown promise in solving partial differential equations (PDEs), including the frequency-domain Helmholtz equation. However, standard training of PINNs using gradient descent (GD) suffers from slow convergence and instability, particularly for high-frequency wavefields. For scattered acoustic wavefield simulation based on Helmholtz equation, we derive a hybrid optimization framework that accelerates training convergence by embedding a least-squares (LS) solver directly into the GD loss function. This formulation enables optimal updates for the linear output layer. Our method is applicable with or without perfectly matched layers (PML), and we provide practical tensor-based implementations for both scenarios. Numerical experiments on benchmark velocity models demonstrate that our approach achieves faster convergence, higher accuracy, and improved stability compared to conventional PINN training. In particular, our results show that the LS-enhanced method converges rapidly even in cases where standard GD-based training fails. The LS solver operates on a small normal matrix, ensuring minimal computational overhead and making the method scalable for large-scale wavefield simulations.

Problem

• Accelerate PINNs convergence for Helmholtz equation
• Improve stability in high-frequency wavefield simulations
• Combine least-squares solver with gradient descent

Innovation

• Hybrid optimization with embedded least-squares solver
• Optimal updates for linear output layer
• Scalable tensor-based implementation for wavefields