ICLR 2025

Published as a conference paper at ICLR 2025
PHYSICS-INFORMED DIFFUSION MODELS
Jan-Hendrik Bastek
Dept. of Mechanical and Process Eng.
ETH Zurich
Zurich, Switzerland
jbastek@ethz.ch
WaiChing Sun
Dept. of Civil Eng. and Eng. Mechanics
Columbia University
New York, NY, USA
wsun@columbia.edu
Dennis M. Kochmann
Dept. of Mechanical and Process Eng.
ETH Zurich
Zurich, Switzerland
dmk@ethz.ch
ABSTRACT
Generative models such as denoising diffusion models are quickly advancing their
ability to approximate highly complex data distributions. They are also increasingly
leveraged in scientific machine learning, where samples from the implied data
distribution are expected to adhere to specific governing equations. We present
a framework that unifies generative modeling and partial differential equation
fulfillment by introducing a first-principle-based loss term that enforces generated
samples to fulfill the underlying physical constraints. Our approach reduces the
residual error by up to two orders of magnitude compared to previous work in a
fluid flow case study and outperforms task-specific frameworks in relevant metrics
for structural topology optimization. We also present numerical evidence that
our extended training objective acts as a natural regularization mechanism against
overfitting. Our framework is simple to implement and versatile in its applicability
for imposing equality and inequality constraints as well as auxiliary optimization
objectives.
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INTRODUCTION
Denoising diffusion models (Sohl-Dickstein et al., 2015) are generative models that have gained
popularity due to their success in learning intricate data distributions across various modalities,
including images (Ho et al., 2020; Nichol et al., 2021; Rombach et al., 2021), videos (Ho et al.,
2022a;b), graphs (Niu et al., 2020; Hoogeboom et al., 2022), text (Li et al., 2022; Wu et al., 2023), and
audio waveforms (Kong et al., 2021). Originally popularized within the image generation community,
their outstandin...