CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problems
This work addresses the problem of physical consistency in generative modeling for scientific machine learning, offering an incremental improvement by extending score-based models to PDE domains with efficient conditioning.
The paper tackles the challenge of enforcing physical laws in generative models for partial differential equations (PDEs), introducing a method that incorporates discretized information into score-based models to generate samples with residuals comparable to conventional PDE solvers, enhancing fidelity. It demonstrates the versatility of these models in tasks like surrogate modeling and probabilistic field reconstruction from sparse measurements.
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.