CoCoGen: Physically-Consistent and Conditioned Score-based Generative
  Models for Forward and Inverse Problems
Christian Jacobsen Yilin Zhuang Karthik Duraisamy
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.