Strictly Constrained Generative Modeling via Split Augmented Langevin Sampling
This addresses the limitation of deploying generative models in scientific and engineering problems where physical constraints are critical, though it appears incremental as it builds on existing Langevin dynamics and diffusion models.
The authors tackled the problem of ensuring physical plausibility in deep generative models by developing a framework for sampling from a target distribution while rigorously satisfying constraints, resulting in improved forecast accuracy and preservation of conserved quantities in diffusion-based data assimilation.
Deep generative models hold great promise for representing complex physical systems, but their deployment is currently limited by the lack of guarantees on the physical plausibility of the generated outputs. Ensuring that known physical constraints are enforced is therefore critical when applying generative models to scientific and engineering problems. We address this limitation by developing a principled framework for sampling from a target distribution while rigorously satisfying physical constraints. Leveraging the variational formulation of Langevin dynamics, we propose Split Augmented Langevin (SAL), a novel primal-dual sampling algorithm that enforces constraints progressively through variable splitting, with convergence guarantees. While the method is developed theoretically for Langevin dynamics, we demonstrate its effective applicability to diffusion models. In particular, we use constrained diffusion models to generate physical fields satisfying energy and mass conservation laws. We apply our method to diffusion-based data assimilation on a complex physical system, where enforcing physical constraints substantially improves both forecast accuracy and the preservation of critical conserved quantities. We also demonstrate the potential of SAL for challenging feasibility problems in optimal control.