# CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problems

arXiv: https://arxiv.org/abs/2312.10527 · HTML: https://mlsanity.com/paper/2312.10527/

## Key facts

- Cited by 56 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 39 of its 55 references that are in the map.
- First posted on arXiv 16 Dec 2023.

## 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, 30 of 39)

- [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)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Adding Conditional Control to Text-to-Image Diffusion Models](https://mlsanity.com/paper/2302.05543/) (2023)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [GLIDE: Towards Photorealistic Image Generation and Editing with Text-Guided Diffusion Models](https://mlsanity.com/paper/2112.10741/) (2021)
- [DreamBooth: Fine Tuning Text-to-Image Diffusion Models for Subject-Driven Generation](https://mlsanity.com/paper/2208.12242/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [RePaint: Inpainting using Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.09865/) (2022)
- [Palette: Image-to-Image Diffusion Models](https://mlsanity.com/paper/2111.05826/) (2021)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Diffusion-LM Improves Controllable Text Generation](https://mlsanity.com/paper/2205.14217/) (2022)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusion](https://mlsanity.com/paper/2310.02279/) (2023)
- [DiffuSeq: Sequence to Sequence Text Generation with Diffusion Models](https://mlsanity.com/paper/2210.08933/) (2022)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Open-Vocabulary Panoptic Segmentation with Text-to-Image Diffusion Models](https://mlsanity.com/paper/2303.04803/) (2023)
- [Score-based diffusion models for accelerated MRI](https://mlsanity.com/paper/2110.05243/) (2021)
- [MADE: Masked Autoencoder for Distribution Estimation](https://mlsanity.com/paper/1502.03509/) (2015)
- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)

## Built on by (most cited first, 30 of 56)

- [DiffusionPDE: Generative PDE-Solving Under Partial Observation](https://mlsanity.com/paper/2406.17763/) (2024)
- [On conditional diffusion models for PDE simulations](https://mlsanity.com/paper/2410.16415/) (2024)
- [Guided Diffusion Sampling on Function Spaces with Applications to PDEs](https://mlsanity.com/paper/2505.17004/) (2025)
- [Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints](https://mlsanity.com/paper/2506.08604/) (2025)
- [Generative Physical AI in Vision: A Survey](https://mlsanity.com/paper/2501.10928/) (2025)
- [FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modeling](https://mlsanity.com/paper/2506.07902/) (2025)
- [Generative Learning of the Solution of Parametric Partial Differential Equations Using Guided Diffusion Models and Virtual Observations](https://mlsanity.com/paper/2408.00157/) (2024)
- [Spatially-Aware Diffusion Models with Cross-Attention for Global Field Reconstruction with Sparse Observations](https://mlsanity.com/paper/2409.00230/) (2024)
- [Source-Guided Flow Matching](https://mlsanity.com/paper/2508.14807/) (2025)
- [Data-Driven Stochastic Closure Modeling via Conditional Diffusion Model and Neural Operator](https://mlsanity.com/paper/2408.02965/) (2024)
- [CFO: Learning Continuous-Time PDE Dynamics via Flow-Matched Neural Operators](https://mlsanity.com/paper/2512.05297/) (2025)
- [Probabilistic operator learning: generative modeling and uncertainty quantification for foundation models of differential equations](https://mlsanity.com/paper/2509.05186/) (2025)
- [Decoupled Diffusion Sampling for Inverse Problems on Function Spaces](https://mlsanity.com/paper/2601.23280/) (2026)
- [Unifying and extending Diffusion Models through PDEs for solving Inverse Problems](https://mlsanity.com/paper/2504.07437/) (2025)
- [Generative Learning for Forecasting the Dynamics of Complex Systems](https://mlsanity.com/paper/2402.17157/) (2024)
- [Conditional score-based diffusion models for solving inverse problems in mechanics](https://mlsanity.com/paper/2406.13154/) (2024)
- [Efficient Flow Matching using Latent Variables](https://mlsanity.com/paper/2505.04486/) (2025)
- [D-Flow SGLD: Source-Space Posterior Sampling for Scientific Inverse Problems with Flow Matching](https://mlsanity.com/paper/2602.21469/) (2026)
- [Physics-informed diffusion models in spectral space](https://mlsanity.com/paper/2602.09708/) (2026)
- [Diffusion models as probabilistic neural operators for recovering unobserved states of dynamical systems](https://mlsanity.com/paper/2405.07097/) (2024)
- [Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Models](https://mlsanity.com/paper/2506.20771/) (2025)
- [Enforcing governing equation constraints in neural PDE solvers via training-free projections](https://mlsanity.com/paper/2511.17258/) (2025)
- [PAD-Hand: Physics-Aware Diffusion for Hand Motion Recovery](https://mlsanity.com/paper/2603.26068/) (2026)
- [PRISMA: Improving the Accuracy-Latency Frontier of Diffusion-based PDE Solvers Using Physics-Informed Spectral Attention](https://mlsanity.com/paper/2512.01370/) (2025)
- [Adaptive Diffusion Posterior Sampling for Data and Model Fusion of Complex Nonlinear Dynamical Systems](https://mlsanity.com/paper/2603.12635/) (2026)
- [Autoregressive regularized score-based diffusion models for multi-scenarios fluid flow prediction](https://mlsanity.com/paper/2505.24145/) (2025)
- [Physics-Informed Diffusion Models for Unsupervised Anomaly Detection in Multivariate Time Series](https://mlsanity.com/paper/2508.11528/) (2025)
- [PIRF: Physics-Informed Reward Fine-Tuning for Diffusion Models](https://mlsanity.com/paper/2509.20570/) (2025)
- [PHDME: Physics-Informed Diffusion Models without Explicit Governing Equations](https://mlsanity.com/paper/2601.21234/) (2026)
- [PILD: Physics-Informed Learning via Diffusion](https://mlsanity.com/paper/2601.21284/) (2026)

## Abstract (arXiv)

Recent advances in generative artificial intelligence have had a significant impact on diverse domains spanning computer vision, natural language processing, and drug discovery. This work extends the reach of generative models into physical problem domains, particularly addressing the efficient enforcement of physical laws and conditioning for forward and inverse problems involving partial differential equations (PDEs). Our work introduces two key contributions: firstly, we present an efficient approach to promote consistency with the underlying PDE. By incorporating discretized information into score-based generative models, our method generates samples closely aligned with the true data distribution, showcasing residuals comparable to data generated through conventional PDE solvers, significantly enhancing fidelity. Secondly, we showcase the potential and versatility of score-based generative models in various physics tasks, specifically highlighting surrogate modeling as well as probabilistic field reconstruction and inversion from sparse measurements. A robust foundation is laid by designing unconditional score-based generative models that utilize reversible probability flow ordinary differential equations. Leveraging conditional models that require minimal training, we illustrate their flexibility when combined with a frozen unconditional model. These conditional models generate PDE solutions by incorporating parameters, macroscopic quantities, or partial field measurements as guidance. The results illustrate the inherent flexibility of score-based generative models and explore the synergy between unconditional score-based generative models and the present physically-consistent sampling approach, emphasizing the power and flexibility in solving for and inverting physical fields governed by differential equations, and in other scientific machine learning tasks.