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

2025-12-26

AI Summary

For PDE-constrained inverse problems involving high-dimensional, non-uniform parameter fields (e.g., subsurface permeability estimation), severe ill-posedness and gradient divergence during optimization arise under sparse observational data. This paper proposes the first end-to-end implicit inversion framework: it couples a pre-trained latent diffusion model (LDM) with a differentiable finite-volume PDE solver, performing inversion entirely in a low-dimensional latent space. Gradient propagation is achieved via adjoint sensitivity analysis combined with reverse-mode automatic differentiation, ensuring exact gradient computation. The latent-parameterization inherently suppresses ill-posed degrees of freedom while preserving sharp material interfaces. Experiments demonstrate significant improvements over physics-informed neural networks (PINNs) and physics-embedded VAEs—particularly in reconstruction accuracy, numerical stability, and robustness to initialization—and enable precise recovery of discontinuous material boundaries.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Diffusion Models for Vision
• Machine Learning: Deep Generative Models & Autoencoders

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized 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.

Problem

• Reconstructs heterogeneous subsurface parameters from sparse observations.
• Improves stability and accuracy in PDE-constrained inverse problems.
• Integrates latent diffusion prior with differentiable numerical solver.

Innovation

• Latent diffusion prior coupled with differentiable solver
• Inversion performed directly in latent space
• Adjoint gradients combined with automatic differentiation