# PILD: Physics-Informed Learning via Diffusion

arXiv: https://arxiv.org/abs/2601.21284 · HTML: https://mlsanity.com/paper/2601.21284/

## 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 44 of its 70 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## 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 44)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [Structured Denoising Diffusion Models in Discrete State-Spaces](https://mlsanity.com/paper/2107.03006/) (2021)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Diffusion Models in Vision: A Survey](https://mlsanity.com/paper/2209.04747/) (2022)
- [Representation Alignment for Generation: Training Diffusion Transformers Is Easier Than You Think](https://mlsanity.com/paper/2410.06940/) (2024)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [DiffusionDet: Diffusion Model for Object Detection](https://mlsanity.com/paper/2211.09788/) (2022)
- [Efficient Diffusion Training via Min-SNR Weighting Strategy](https://mlsanity.com/paper/2303.09556/) (2023)
- [Diffusion-based Time Series Imputation and Forecasting with Structured State Space Models](https://mlsanity.com/paper/2208.09399/) (2022)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Adversarial purification with Score-based generative models](https://mlsanity.com/paper/2106.06041/) (2021)
- [Diffusion Models for Image Restoration and Enhancement: A Comprehensive Survey](https://mlsanity.com/paper/2308.09388/) (2023)
- [Exploring Chemical Space with Score-based Out-of-distribution Generation](https://mlsanity.com/paper/2206.07632/) (2022)
- [Understanding Hallucinations in Diffusion Models through Mode Interpolation](https://mlsanity.com/paper/2406.09358/) (2024)
- [Physics-Informed Diffusion Models](https://mlsanity.com/paper/2403.14404/) (2024)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Residual Denoising Diffusion Models](https://mlsanity.com/paper/2308.13712/) (2023)
- [A Diffusion Model Framework for Unsupervised Neural Combinatorial Optimization](https://mlsanity.com/paper/2406.01661/) (2024)
- [CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problems](https://mlsanity.com/paper/2312.10527/) (2023)
- [Neural Network Diffusion](https://mlsanity.com/paper/2402.13144/) (2024)
- [REPA Works Until It Doesn't: Early-Stopped, Holistic Alignment Supercharges Diffusion Training](https://mlsanity.com/paper/2505.16792/) (2025)

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

- [When to Lock Attention: Training-Free KV Control in Video Diffusion](https://mlsanity.com/paper/2603.09657/) (2026)
- [Self-Augmented Diffusion Guidance for Physics-Informed Generation](https://mlsanity.com/paper/2608.26748/) (2026)

## Abstract (arXiv)

Diffusion models have emerged as powerful generative tools for modeling complex data distributions, yet their purely data-driven nature limits applicability in engineering and scientific problems where physical laws must be respected. This paper proposes Physics-Informed Learning via Diffusion (PILD), a framework that unifies diffusion modeling and physical constraints through a probabilistic residual formulation with a virtual residual observation sampled from a Laplace distribution. To make this formulation practical under noisy diffusion states, we introduce a Jensen-gap-aware adaptive residual scale, which reduces the bias induced by residual likelihood marginalization. Additionally, we develop a physics-conditional alignment mechanism for conditional tasks that encourages intermediate latent representations to remain consistent with the observation conditions during denoising. The proposed framework is concise, modular, and broadly applicable to problems governed by ordinary differential equations, partial differential equations, as well as algebraic equations or inequality constraints. Extensive experiments across engineering and scientific tasks show that PILD improves physical fidelity and predictive accuracy over representative physics-informed and diffusion-based baselines.