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Recent advances in generative artificial intelligence have had a significantimpact on diverse domains spanning computer vision, natural languageprocessing, and drug discovery. This work extends the reach of generativemodels into physical problem domains, particularly addressing the efficientenforcement of physical laws and conditioning for forward and inverse problemsinvolving partial differential equations (PDEs). Our work introduces two keycontributions: firstly, we present an efficient approach to promote consistencywith the underlying PDE. By incorporating discretized information intoscore-based generative models, our method generates samples closely alignedwith the true data distribution, showcasing residuals comparable to datagenerated through conventional PDE solvers, significantly enhancing fidelity.Secondly, we showcase the potential and versatility of score-based generativemodels in various physics tasks, specifically highlighting surrogate modelingas well as probabilistic field reconstruction and inversion from sparsemeasurements. A robust foundation is laid by designing unconditionalscore-based generative models that utilize reversible probability flow ordinarydifferential equations. Leveraging conditional models that require minimaltraining, we illustrate their flexibility when combined with a frozenunconditional model. These conditional models generate PDE solutions byincorporating parameters, macroscopic quantities, or partial field measurementsas guidance. The results illustrate the inherent flexibility of score-basedgenerative models and explore the synergy between unconditional score-basedgenerative models and the present physically-consistent sampling approach,emphasizing the power and flexibility in solving for and inverting physicalfields governed by differential equations, and in other scientific machinelearning tasks.

CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problems