Latent Diffusion Posterior Sampling with Surrogate Likelihood Guidance for PDE Inverse Problems

2026-06-25 · Source: Machine Learning · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Scientific Machine Learning · Depth: Expert, quick

Summary

Latent-space diffusion posterior sampling (L-DPS) is an approximate Bayesian framework for high-dimensional inverse problems governed by partial differential equations (PDEs). This method addresses challenges such as implicit sample-based priors, high-dimensional spatially distributed parameters, and the high cost of repeated forward-model evaluations. L-DPS combines a variational autoencoder (VAE) to map parameter fields to a latent space, an unconditional latent diffusion model for implicit prior score learning, and diffusion posterior sampling (DPS) for likelihood-based guidance. A differentiable neural surrogate evaluates the likelihood gradient via a decoder-surrogate composition, avoiding full numerical PDE solver calls. Evaluated on an inverse Darcy flow problem, L-DPS produced accurate, robust inverse solutions, reduced inference cost versus full-space DPS, and outperformed amortized inverse baselines like conditional latent diffusion and inverse FNO in sparse and noisy regimes. The study also examined mixed-prior generalization and sensitivity to surrogate forward-model error.

Key takeaway

For research scientists and ML engineers tackling high-dimensional PDE inverse problems, L-DPS offers a robust and computationally efficient Bayesian inversion approach. You should consider integrating latent diffusion models and neural surrogates to manage implicit priors and reduce expensive forward-model evaluations. This method improves accuracy and robustness, especially with sparse and noisy observations, making it a strong candidate for complex scientific and engineering simulations.

Key insights

L-DPS uses latent diffusion and a neural surrogate for efficient, accurate Bayesian inversion of PDEs.

Principles

• Implicit priors can be learned via latent diffusion.
• Surrogate models reduce computational cost in inversion.
• Bayesian frameworks enhance robustness in noisy data.

Method

L-DPS maps parameter fields to a latent space via VAE, learns an implicit prior score with a latent diffusion model, and uses DPS with a differentiable neural surrogate for likelihood gradient evaluation.

In practice

• Apply L-DPS for high-dimensional PDE inverse problems.
• Use VAEs to reduce parameter dimensionality.
• Integrate neural surrogates to accelerate PDE solvers.

Topics

• Latent Diffusion Posterior Sampling
• PDE Inverse Problems
• Bayesian Inference
• Neural Surrogate Models
• Variational Autoencoders
• Darcy Flow

Best for: AI Scientist, Machine Learning Engineer, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by Machine Learning.