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

TL;DR

This work tackles ill-posed, high-dimensional PDE-constrained inverse problems in subsurface flow by embedding a learned latent diffusion prior within a differentiable forward solver, enabling end-to-end gradient-based inversion in a compact latent space. By encoding conductivity fields into a geologically informed latent manifold and solving Darcy flow with a differentiable finite-volume operator, LD-DIM achieves stable optimization and preserves sharp interfaces, outperforming PINN and VAE baselines. The method combines a two-stage latent diffusion model (VAE + diffusion) with adjoint-based sensitivities to compute exact gradients through the discretization, demonstrated on Gaussian and bimaterial conductivity fields with varying data sparsity. The results show improved reconstruction accuracy, reduced sensitivity to initialization, and maintained physical consistency, indicating strong potential for scalable, physics-constrained inverse modeling in complex geophysical systems.

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.

Figures (16)

• Figure 1: Training workflow of the latent diffusion model (LDM). The LDM comprises two key components: (a) a variational autoencoder (VAE) that compresses conductivity fields into low-dimensional latent representations, and (b) a diffusion process performed in the latent space, consisting of a forward diffusion that progressively adds noise and a reverse denoising process that removes noise and reconstructs clean latent representations using a denoising U-Net. Training is conducted in two stages: the VAE is first trained and fixed, followed by training the U-Net model to learn the reverse denoising dynamics in the latent space.
• Figure 2: LD-DIM inverse modeling workflow. The pretrained latent diffusion model maps a low-dimensional latent vector $\hat{\bm{z}}$ into a conductivity field via reverse denoising and decoding. The generated conductivity field is input to a differentiable finite-volume solver that computes the hydraulic head $\hat{h}$, which is compared with observations $h^{*}$ in the objective function. Gradients with respect to latent variable $\hat{\bm{z}}$ are efficiently computed by combining adjoint-based gradients through the finite-volume solver and automatic differentiation through the pre-trained latent diffusion model, enabling end-to-end optimization in the compact latent space.
• Figure 3: Representative samples of Gaussian conductivity fields corresponding to different correlation lengths $\lambda$. Top row (left to right): $\lambda=0.1,\,0.2,\,0.3$. Bottom row (left to right): $\lambda=0.4,\,0.5,\,0.6$. Fields exhibit progressively smoother spatial structure with increased correlation length.
• Figure 4: Representative testing samples of Gaussian conductivity fields $K$ (top) and the corresponding hydraulic head fields $h$ (bottom) for correlation lengths $\lambda = 0.1,\,0.2,\,0.3,$ and $0.4$ (left to right). These test data serve as the ground truth for the inverse experiments.
• Figure 5: Inversion results by LD-DIM for Gaussian conductivity fields with varying correlation lengths. Each row displays (left to right): predicted conductivity field, absolute error in conductivity, and error in hydraulic head. The framework demonstrates improved accuracy for larger correlation lengths due to reduced high-frequency content and enhanced information content in sparse observations.