Rethinking neural network reparameterized full waveform inversion: A Hessian perspective

By seamlessly integrating deep neural networks (DNNs) into the traditional full waveform inversion (FWI) framework, the neural-network-reparameterized FWI (RFWI) naturally introduces implicit regularization, thereby mitigating key challenges in practical applications-such as sensitivity to initial models and susceptibility to local minima. While a variety of advanced RFWI methods have been proposed to enhance inversion performance, profound insights remain lacking in two key questions: (1) How does the DNN improve the robustness of RFWI, (2) How to further improve the performance of RFWI. In response to the first question, the Hessian spectrum is introduced as an analytical tool to reveal the difference in optimization characteristics between RFWI and traditional FWI. Our findings demonstrate that the superior robustness arises from the persistent constraints imposed by DNN on the inversion process, guiding a long-lasting tendency towards the eigenvectors associated with large eigenvalues, which correspond to stable directions dominated by low-frequency information. For the second question, we propose two improvement techniques based on the inherent optimization characteristics of RFWI, targeting convergence or robustness, both of which have the potential to generalize to other RFWI implementations. Focusing on convergence improvement in ideal scenarios, we propose the Adam+L-BFGS hybrid optimizer, which accelerates high-frequency fitting while avoiding local minima. Regarding robustness improvement in challenging scenarios, we propose a Hessian trace-based regularization technique that promotes flatter optimization trajectories, thereby mitigating the overfitting of noise. The experimental results support the effectiveness of our proposed improvement techniques.