Multi-frequency wavefield solutions for variable velocity models using meta-learning enhanced low-rank physics-informed neural network

TL;DR

Meta-LRPINN addresses the challenge of fast and accurate multi-frequency seismic wavefield modeling in variable velocity models by coupling a low-rank, SVD-based PINN with a frequency embedding hypernetwork. The FEH ties input frequency to layer singular values, while meta-learning provides robust initialization for rapid adaptation across velocity models and frequencies; adaptive rank reduction and FEH pruning further reduce computational cost. Numerical experiments show Meta-LRPINN achieves faster convergence and higher accuracy than Meta-PINN and vanilla PINN, with strong generalization to out-of-distribution frequencies. The framework offers a scalable approach for seismic wavefield representation, enabling efficient multi-frequency analyses in complex subsurface settings.

Abstract

Physics-informed neural networks (PINNs) face significant challenges in modeling multi-frequency wavefields in complex velocity models due to their slow convergence, difficulty in representing high-frequency details, and lack of generalization to varying frequencies and velocity scenarios. To address these issues, we propose Meta-LRPINN, a novel framework that combines low-rank parameterization using singular value decomposition (SVD) with meta-learning and frequency embedding. Specifically, we decompose the weights of PINN's hidden layers using SVD and introduce an innovative frequency embedding hypernetwork (FEH) that links input frequencies with the singular values, enabling efficient and frequency-adaptive wavefield representation. Meta-learning is employed to provide robust initialization, improving optimization stability and reducing training time. Additionally, we implement adaptive rank reduction and FEH pruning during the meta-testing phase to further enhance efficiency. Numerical experiments, which are presented on multi-frequency scattered wavefields for different velocity models, demonstrate that Meta-LRPINN achieves much fast convergence speed and much high accuracy compared to baseline methods such as Meta-PINN and vanilla PINN. Also, the proposed framework shows strong generalization to out-of-distribution frequencies while maintaining computational efficiency. These results highlight the potential of our Meta-LRPINN for scalable and adaptable seismic wavefield modeling.

Figures (19)

• Figure 1: Illustration of the proposed LRPINN with frequency embedding. (a) Frequency embedding hypernetwork: This network maps the input frequency ($\textit{freq}$) to frequency-dependent singular values ($\sigma_{l,1}, \sigma_{l,2}, \dots, \sigma_{l,k}$) for each layer through a series of linear layers with sine and GELU activations. These singular values are used to construct diagonal matrices ($\Theta_l$) that dynamically adapt to the input frequency. (b) Low-rank PINN: The inputs (spatial location $\mathbf{x} =(x,z)$ and source location $\mathbf{x}_s =(x_s,z_s)$) are encoded using sine activations, and each layer's weight matrix ($W_l$) is factorized into $U_l \cdot \Theta_l \cdot V_l^\text{T}$, incorporating the frequency-dependent singular values from (a). The network outputs the real ($\delta u_R(\mathbf{x}, \mathbf{x}_s, \omega)$) and imaginary ($\delta u_I(\mathbf{x}, \mathbf{x}_s, \omega)$) components of the scattered wavefield.
• Figure 2: The layered velocity model extracted from the Marmousi model.
• Figure 3: Comparison of physical loss and accuracy curves between our Meta-LRPINN (blue), Meta-PINN (orange), and vanilla PINN (yellow) on the layered velocity model. The rows correspond to different frequencies (top: 3 Hz, middle: 6 Hz, bottom: 12 Hz). For each frequency, the left column shows the physical loss curves, and the right column presents the accuracy curves (measured as MSE against the numerical reference).
• Figure 4: Comparison of the real part of the scattered wavefield solutions at 3 Hz for the layered velocity model. (a) Numerical reference solution. Subsequent rows represent wavefields predicted by Meta-LRPINN, Meta-PINN, and vanilla PINN, respectively. Columns correspond to different training epochs, where the specific epoch numbers are indicated in the top.
• Figure 5: Similar with Figure \ref{['fig4']}, but for the scattered wavefield solutions of 6 Hz.