Physics‐Constrained Variational Autoencoder for Uncertainty Quantification of Full Waveform Inversion

Abstract We propose a variational autoencoder framework to directly assess uncertainties in subsurface models produced by single‐ and multiparameter full waveform inversion (FWI). The new method does not require pretraining on labeled data, thus it significantly reduces computational cost and storage requirements. The framework takes the seismic shot gathers as input and returns a set of possible velocity models that fit the input data. The network has three key components: (a) an encoder that maps the input seismic shot gathers to feature distributions in the latent space, (b) a latent vector sampled directly from feature distributions, and (c) a decoder that maps the sampled latent vector to the velocity model space. We incorporated Deep Image Prior principles, leveraging convolutional layers and LeakyReLU activations to regularize the inversion and improve reconstruction. At each epoch, the reconstructed velocity model is passed to a finite difference partial differential equation solver for forward modeling. We then calculate the data misfit between the input seismic data and the simulated data. Additionally, we compute the FWI gradient by cross‐correlating the adjoint and forward wavefield; the FWI gradient is then back‐propagated to guide the update of the neural networks weights and biases. The injection of physics‐based gradients further constrains the inversion and enhancing convergence. We tested the framework on synthetic acoustic and elastic seismic data sets, inverting models of varying complexity and quantifying their associated uncertainties. The results show accurate mean velocity models with meaningful uncertainty estimates, highlighting the potential of the proposed method for practical FWI applications.