# OCL-PDE: A Generative Framework for PDE Inverse Problems with Observation-Complementary Latents

arXiv: https://arxiv.org/abs/2610.06259 · HTML: https://mlsanity.com/paper/2610.06259/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 48 references that are in the map.
- First posted on arXiv 5 Oct 2026.

## Builds on (most cited first, 6 of 6)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [A unified physics-informed generative operator framework for general inverse problems](https://mlsanity.com/paper/2511.03241/) (2025)
- [Bayesian Inverse Problems Meet Flow Matching: Efficient and Flexible Inference via Transformers](https://mlsanity.com/paper/2503.01375/) (2025)

## Abstract (arXiv)

Partial differential equation (PDE) inverse problems are often ill-posed, making fine-scale details difficult to recover. We address this problem by introducing a learned observation-complementary latent representation that preserves reconstruction-relevant information and is combined with the observation to reconstruct the unknown field. Building on this representation, we propose OCL-PDE, a generative framework that encourages the observation to guide large-scale structure and the latent to supply complementary fine-scale details. OCL-PDE is built on a physics-aware autoencoder (AE) and conditional Flow Matching, supporting inverse reconstruction as well as forward PDE prediction. Experiments demonstrate improved reconstruction accuracy and fine-detail recovery compared with the evaluated baselines.