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

arXiv: https://arxiv.org/abs/2512.22421 · HTML: https://mlsanity.com/paper/2512.22421/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 21 of its 66 references that are in the map.
- First posted on arXiv 27 Dec 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 21 of 21)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Demystifying MMD GANs](https://mlsanity.com/paper/1801.01401/) (2018)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [A Differentiable Programming System to Bridge Machine Learning and Scientific Computing](https://mlsanity.com/paper/1907.07587/) (2019)
- [Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport](https://mlsanity.com/paper/1912.02968/) (2019)
- [JAX-FEM: A differentiable GPU-accelerated 3D finite element solver for automatic inverse design and mechanistic data science](https://mlsanity.com/paper/2212.00964/) (2022)
- [JAX-FLUIDS: A fully-differentiable high-order computational fluid dynamics solver for compressible two-phase flows](https://mlsanity.com/paper/2203.13760/) (2022)
- [Training-image based geostatistical inversion using a spatial generative adversarial neural network](https://mlsanity.com/paper/1708.04975/) (2017)
- [A prior regularized full waveform inversion using generative diffusion models](https://mlsanity.com/paper/2306.12776/) (2023)
- [Likelihood-informed dimension reduction for nonlinear inverse problems](https://mlsanity.com/paper/1403.4680/) (2014)
- [Inversion using a new low-dimensional representation of complex binary geological media based on a deep neural network](https://mlsanity.com/paper/1710.09196/) (2017)
- [Differentiable Programming for Earth System Modeling](https://mlsanity.com/paper/2208.13825/) (2022)
- [Parametric generation of conditional geological realizations using generative neural networks](https://mlsanity.com/paper/1807.05207/) (2018)
- [Latent Diffusion Model for Conditional Reservoir Facies Generation](https://mlsanity.com/paper/2311.01968/) (2023)
- [JAX-MPM: A Learning-Augmented Differentiable Meshfree Framework for GPU-Accelerated Lagrangian Simulation and Geophysical Inverse Modeling](https://mlsanity.com/paper/2507.04192/) (2025)
- [Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanics](https://mlsanity.com/paper/2311.12915/) (2023)
- [Gaussian process regression and conditional Karhunen-Lo'eve models for data assimilation in inverse problems](https://mlsanity.com/paper/2301.11279/) (2023)
- [3D latent diffusion models for parameterizing and history matching multiscenario facies systems](https://mlsanity.com/paper/2508.16621/) (2025)

## Built on by (most cited first, 2 of 2)

- [GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Media](https://mlsanity.com/paper/2602.14642/) (2026)
- [Data assimilation for subsurface flow using latent diffusion model parameterization: performance of ensemble-Kalman and Monte Carlo techniques](https://mlsanity.com/paper/2606.11140/) (2026)

## Abstract (arXiv)

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.