Differentiable Inverse Modeling with Physics-Constrained Latent Diffusion for Heterogeneous Subsurface Parameter Fields

TL;DR

This paper introduces LD-DIM, a novel differentiable inverse modeling method that combines latent diffusion priors with PDE solvers to accurately reconstruct complex subsurface parameters from sparse data.

Contribution

The paper proposes a new framework integrating latent diffusion models with differentiable PDE solvers for stable, high-quality inverse solutions in high-dimensional settings.

Findings

Improved numerical stability over PINNs and VAEs.

Accurate reconstruction of sharp interfaces in heterogeneous fields.

Enhanced robustness to initialization and sparse data.

Abstract

We present a latent diffusion-based differentiable inversion method (LD-DIM) for PDE-constrained inverse problems involving high-dimensional spatially distributed coefficients. LD-DIM couples a pretrained latent diffusion prior with an end-to-end differentiable numerical solver to reconstruct unknown heterogeneous parameter fields in a low-dimensional nonlinear manifold, improving numerical conditioning and enabling stable gradient-based optimization under sparse observations. The proposed framework integrates a latent diffusion model (LDM), trained in a compact latent space, with a differentiable finite-volume discretization of the forward PDE. Sensitivities are propagated through the discretization using adjoint-based gradients combined with reverse-mode automatic differentiation. Inversion is performed directly in latent space, which implicitly suppresses ill-conditioned degrees of freedom while preserving dominant structural modes, including sharp material interfaces. The effectiveness of LD-DIM is demonstrated using a representative inverse problem for flow in porous media, where heterogeneous conductivity fields are reconstructed from spatially sparse hydraulic head measurements. Numerical experiments assess convergence behavior and reconstruction quality for both Gaussian random fields and bimaterial coefficient distributions. The results show that LD-DIM achieves consistently improved numerical stability and reconstruction accuracy of both parameter fields and corresponding PDE solutions compared with physics-informed neural networks (PINNs) and physics-embedded variational autoencoder (VAE) baselines, while maintaining sharp discontinuities and reducing sensitivity to initialization.