SPAMoE: Spectrum-Aware Hybrid Operator Framework for Full-Waveform Inversion

Full-waveform inversion (FWI) is a powerful approach for reconstructing subsurface velocity models, but its practical performance is limited by ill-conditioning, sensitivity to the update direction, and the high cost of curvature-based methods. In this project, we investigate whether a neural operator trained on reverse-time-migration (RTM) images can provide a useful learned preconditioner for FWI. We compare projected FWI against neural-preconditioned FWI using either a standard Fourier neural operator (FNO) or a Jacobian-informed FNO (JI-FNO). The neural preconditioner applies the normal operator of the learned RTM map to modify the true Devito FWI gradient. Our experiments show that the JI-FNO preconditioner achieves the lowest model error while maintaining data-residual decay comparable to baseline projected FWI. In contrast, the standard FNO preconditioner is unstable and gives worse model recovery, despite using the same neural-preconditioning framework. We also identify a failure mode of a nonlinear neural Gauss Newton-type inner solve: directly matching the neural RTM output to the FWI gradient creates an objective-space mismatch and does not yield a reliable descent direction. These results suggest that forward RTM-image prediction alone is insufficient for learned preconditioning; accurate learned derivatives and objective-consistent use of the neural operator are critical for producing useful inversion updates.

Full-waveform inversion (FWI) is a high-resolution seismic imaging method that estimates subsurface velocity by matching simulated and recorded waveforms. However, FWI is highly nonlinear, prone to cycle skipping, and sensitive to noise, particularly when low frequencies are missing or the initial model is poor, leading to failures under imperfect acquisition. Diffusion-regularized FWI introduces generative priors to encourage geologically realistic models, but these priors typically require costly offline pretraining and can deteriorate under distribution shift. Moreover, they assume Gaussian initialization and a fixed noise schedule, in which it is unclear how to map a deterministic FWI iterate and its starting model to a well-defined diffusion time or noise level. To address these limitations, we introduce Self-Flow-Matching assisted Full-Waveform Inversion (SFM-FWI), a physics-driven framework that eliminates the need for large-scale offline pretraining while avoiding the noise-level alignment ambiguity. SFM-FWI leverages flow matching to learn a transport field without assuming Gaussian initialization or a predefined noise schedule, so the initial model can be used directly as the starting point of the dynamics. Our approach trains a single flow network online using the governing physics and observed data. At each outer iteration, we build an interpolated model and update the flow by backpropagating the FWI data misfit, providing self-supervision without external training pairs. Experiments on challenging synthetic benchmarks show that SFM-FWI delivers more accurate reconstructions, greater noise robustness, and more stable convergence than standard FWI and pretraining-free regularization methods.

Full-waveform inversion (FWI) estimates physical parameters in the wave equation from limited measurements and has been widely applied in geophysical exploration, medical imaging, and non-destructive testing. Conventional FWI methods are limited by their notorious sensitivity to the accuracy of the initial models. Recent progress in continuous representation FWI (CR-FWI) demonstrates that representing parameter models with a coordinate-based neural network, such as implicit neural representation (INR), can mitigate the dependence on initial models. However, its underlying mechanism remains unclear, and INR-based FWI shows slower high-frequency convergence. In this work, we investigate the general CR-FWI framework and develop a unified theoretical understanding by extending the neural tangent kernel (NTK) for FWI to establish a wave-based NTK framework. Unlike standard NTK, our analysis reveals that wave-based NTK is not constant, both at initialization and during training, due to the inherent nonlinearity of FWI. We further show that the eigenvalue decay behavior of the wave-based NTK can explain why CR-FWI alleviates the dependency on initial models and shows slower high-frequency convergence. Building on these insights, we propose several CR-FWI methods with tailored eigenvalue decay properties for FWI, including a novel hybrid representation combining INR and multi-resolution grid (termed IG-FWI) that achieves a more balanced trade-off between robustness and high-frequency convergence rate. Applications in geophysical exploration on Marmousi, 2D SEG/EAGE Salt and Overthrust, 2004 BP model, and the more realistic 2014 Chevron models show the superior performance of our proposed methods compared to conventional FWI and existing INR-based FWI methods.