LATENT DIFFUSION POSTERIOR SAMPLING WITH SURROGATE LIKELIHOOD GUIDANCE FOR PDE INVERSE PROBLEMS

Solving inverse problems governed by partial differential equations (PDEs) is computationally expensive, especially when the parameter fields are high-dimensional and the prior distribution is only available as a set of samples. The authors seek a method to perform Bayesian inference that avoids the high cost of repeated full-model PDE simulations while handling complex, non-Gaussian priors.

The proposed L-DPS framework addresses these challenges through a multi-stage approach:

1. Dimensionality Reduction: A Variational Autoencoder (VAE) maps high-dimensional parameter fields into a lower-dimensional latent space, preserving spatial structure.
2. Prior Modeling: An unconditional latent diffusion model is trained to learn the score function of the prior distribution within this latent space.
3. Likelihood Guidance: Instead of running the full numerical PDE solver, the method uses a pre-trained differentiable neural surrogate (e.g., Fourier Neural Operator) to approximate the forward model. This allows for efficient gradient-based likelihood evaluation during the reverse diffusion process.

L-DPS successfully produces accurate and robust inverse solutions for Darcy flow problems, even with sparse and noisy observations. By operating in the latent space, the method significantly reduces inference costs compared to full-space diffusion posterior sampling. Furthermore, the study demonstrates that L-DPS outperforms existing amortized baselines (like conditional latent diffusion and inverse FNO) and provides superior results compared to Gaussian-approximated methods (KLE-MAP), particularly when dealing with non-Gaussian parameter fields. The authors also show that while a 'foundational' mixed-prior model is viable, prior-specific models remain the most accurate.