First, I propose a deep learning-based approach for approximating Hessian-related effects in FWI. By training a neural network to map remigrated gradient-domain quantities to improved gradient updates, the method acts as a Hessian-inspired deblurring and illumination-compensation operator. Numerical examples show that the resulting updates improve convergence behavior and produce higher-quality inverted models compared with purely gradient-based updates. 

Second, I introduce a learned time-shift objective for FWI. The network estimates time shifts between modeled and observed seismic traces, and I incorporate these estimates into the adjoint-state framework to construct an adjoint source that emphasizes traveltime discrepancies rather than sample-by-sample amplitude differences. This formulation mitigates cycle skipping while remaining computationally efficient relative to conventional alignment-based approaches. 

Third, I propose a feature-space objective function based on a pre-trained seismic foundation model. By comparing modeled and observed seismic data in the latent space of SeisLM, I replace conventional waveform-domain comparison with a learned representation-space misfit. Numerical experiments show that this objective improves early-stage inversion robustness and guides FWI toward more geologically coherent velocity models. 

Finally, I critically evaluate supervised learning for modifying encoded multi-source gradients. I show that neural networks can reduce visible crosstalk artifacts in encoded gradients, but that improved gradient appearance does not necessarily translate into better inversion performance. This negative result is important because it demonstrates that learning-based corrections must be judged by their effect on optimization behavior and model recovery, not only by intermediate visual quality. 

Overall, my results show that deep learning can be effective in FWI when it is embedded as a targeted, interpretable component within the physics-based inversion loop. The dissertation contributes both positive methods for Hessian approximation and misfit-function design, and a critical analysis of the limitations of learning-based gradient manipulation in scalable multi-source inversion.