Least-squares-embedded optimization for accelerated convergence of PINNs in high-frequency acoustic wavefield simulations

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 in seismic scales. For scattered acoustic wavefield simulation based on the Helmholtz equation, we derive a hybrid optimization framework by embedding a least-squares (LS) solver directly into the GD loss function. This approach enables optimal updates for the linear output layer and mitigates the spectral bias of neural networks, making it especially effective for high-frequency problems. We provide practical, tensor-based implementations for Helmholtz formulations with and without perfectly matched layers (PML). Numerical experiments on benchmark velocity models demonstrate that the proposed method achieves faster convergence, higher accuracy, improved stability, and greater robustness to hyperparameter selection 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. Since the LS solver operates on small matrices embedded within the loss, it introduces minimal computational overhead and remains scalable for large-scale wavefield simulations. • Accelerates training for high-frequency Helmholtz wavefield simulations. • More accurate and more stable solutions. • Mitigates the low-frequency bias of neural networks. • Performs well even when the training data are limited.