On the role of memorization in learned priors for geophysical inverse problems

• URL: http://arxiv.org/abs/2603.19629v1
• Date: Fri, 20 Mar 2026 04:22:26 GMT
• Title: On the role of memorization in learned priors for geophysical inverse problems
• Abstract summary: memorization yields a Gaussian mixture prior in closed form for diffusion models.<n>We show the consequences of memorization through diffusion posterior sampling for full waveform inversion.
• Score: 2.5405233238972564
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Learned priors based on deep generative models offer data-driven regularization for seismic inversion, but training them requires a dataset of representative subsurface models -- a resource that is inherently scarce in geoscience applications. Since the training objective of most generative models can be cast as maximum likelihood on a finite dataset, any such model risks converging to the empirical distribution -- effectively memorizing the training examples rather than learning the underlying geological distribution. We show that the posterior under such a memorized prior reduces to a reweighted empirical distribution -- i.e., a likelihood-weighted lookup among the stored training examples. For diffusion models specifically, memorization yields a Gaussian mixture prior in closed form, and linearizing the forward operator around each training example gives a Gaussian mixture posterior whose components have widths and shifts governed by the local Jacobian. We validate these predictions on a stylized inverse problem and demonstrate the consequences of memorization through diffusion posterior sampling for full waveform inversion.

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