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Autoregressive next-step prediction models have become the de-facto standard for building data-driven neural solvers to forecast time-dependent partial differential equations (PDEs). Denoise training that is closely related to diffusion…

Machine Learning · Computer Science 2025-03-31 Zijie Li , Anthony Zhou , Amir Barati Farimani

This work presents a physics-conditioned latent diffusion model tailored for dynamical downscaling of atmospheric data, with a focus on reconstructing high-resolution 2-m temperature fields. Building upon a pre-existing diffusion…

Machine Learning · Computer Science 2025-10-29 Paul Rosu , Muchang Bahng , Erick Jiang , Rico Zhu , Vahid Tarokh

Neural networks are one tool for approximating non-linear differential equations used in scientific computing tasks such as surrogate modeling, real-time predictions, and optimal control. PDE foundation models utilize neural networks to…

Machine Learning · Computer Science 2025-02-11 Elisa Negrini , Yuxuan Liu , Liu Yang , Stanley J. Osher , Hayden Schaeffer

We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric partial differential equations (PDEs) conditioned on partial observations, which includes,…

Machine Learning · Computer Science 2026-02-11 Davide Gallon , Philippe von Wurstemberger , Patrick Cheridito , Arnulf Jentzen

Diffusion models have recently emerged as powerful stochastic frameworks for high-dimensional inference and generation. However, existing applications to partial differential equations (PDEs) predominantly rely on physics-informed training…

Numerical Analysis · Mathematics 2026-04-03 Yi Bing , Liu Jia , Fu Jinyang , Peng Xiang

We explore training deep neural network models in conjunction with physics simulations via partial differential equations (PDEs), using the simulated degrees of freedom as latent space for a neural network. In contrast to previous work,…

Machine Learning · Computer Science 2023-10-04 Chloe Paliard , Nils Thuerey , Kiwon Um

Scientific measurements are often bottlenecked by suboptimal conditions, whether that be noise, incomplete spatial coverage, or limited resolution, rendering accurate field reconstruction a difficult task. We introduce LatentPDE, a latent…

Machine Learning · Computer Science 2026-04-28 Valerie Tsao , Nathaniel Chaney , Manolis Veveakis

Solving partial differential equations (PDEs) on fine spatio-temporal scales for high-fidelity solutions is critical for numerous scientific breakthroughs. Yet, this process can be prohibitively expensive, owing to the inherent complexities…

Numerical Analysis · Mathematics 2024-04-09 Yulong Lu , Wuzhe Xu

Industrial design evaluation often relies on high-fidelity simulations of governing partial differential equations (PDEs). While accurate, these simulations are computationally expensive, making dense exploration of design spaces…

Machine Learning · Computer Science 2025-10-01 Zhizhou Zhang , Youjia Wu , Kaixuan Zhang , Yanjia Wang

Partial differential equations (PDEs) govern nearly every physical process in science and engineering, yet solving them at scale remains prohibitively expensive. Generative AI has transformed language, vision, and protein science, but…

Machine Learning · Computer Science 2026-04-10 Yilong Dai , Shengyu Chen , Xiaowei Jia , Runlong Yu

The steep computational cost of diffusion models at inference hinders their use as fast physics emulators. In the context of image and video generation, this computational drawback has been addressed by generating in the latent space of an…

Machine Learning · Computer Science 2025-11-04 François Rozet , Ruben Ohana , Michael McCabe , Gilles Louppe , François Lanusse , Shirley Ho

Diffusion-based models have demonstrated impressive accuracy and generalization in solving partial differential equations (PDEs). However, they still face significant limitations, such as high sampling costs and insufficient physical…

Machine Learning · Computer Science 2026-02-04 Cindy Xiangrui Kong , Yueqi Wang , Haoyang Zheng , Weijian Luo , Guang Lin

In many mechanistic medical, biological, physical and engineered spatiotemporal dynamic models the numerical solution of partial differential equations (PDEs) can make simulations impractically slow. Biological models require the…

Soft Condensed Matter · Physics 2021-02-11 J. Quetzalcóatl Toledo-Marín , Geoffrey Fox , James P. Sluka , James A. Glazier

Diffusion models have gained prominence in generating high-quality sequences of text. Nevertheless, current approaches predominantly represent discrete text within a continuous diffusion space, which incurs substantial computational…

Machine Learning · Computer Science 2023-10-17 Shansan Gong , Mukai Li , Jiangtao Feng , Zhiyong Wu , Lingpeng Kong

Neural solvers for partial differential equations (PDEs) have great potential to generate fast and accurate physics solutions, yet their practicality is currently limited by their generalizability. PDEs evolve over broad scales and exhibit…

Machine Learning · Computer Science 2024-12-06 Anthony Zhou , Amir Barati Farimani

Solving time-dependent Partial Differential Equations (PDEs) using a densely discretized spatial domain is a fundamental problem in various scientific and engineering disciplines, including modeling climate phenomena and fluid dynamics.…

Machine Learning · Computer Science 2025-10-24 Jan Hagnberger , Daniel Musekamp , Mathias Niepert

Neural networks have shown great potential in accelerating the solution of partial differential equations (PDEs). Recently, there has been a growing interest in introducing physics constraints into training neural PDE solvers to reduce the…

Machine Learning · Computer Science 2023-05-30 Xinquan Huang , Wenlei Shi , Qi Meng , Yue Wang , Xiaotian Gao , Jia Zhang , Tie-Yan Liu

By decomposing the image formation process into a sequential application of denoising autoencoders, diffusion models (DMs) achieve state-of-the-art synthesis results on image data and beyond. Additionally, their formulation allows for a…

Computer Vision and Pattern Recognition · Computer Science 2022-04-14 Robin Rombach , Andreas Blattmann , Dominik Lorenz , Patrick Esser , Björn Ommer

Neural networks have shown promising potential in accelerating the numerical simulation of systems governed by partial differential equations (PDEs). Different from many existing neural network surrogates operating on high-dimensional…

Machine Learning · Computer Science 2025-01-09 Zijie Li , Saurabh Patil , Francis Ogoke , Dule Shu , Wilson Zhen , Michael Schneier , John R. Buchanan, , Amir Barati Farimani

Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable…

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