Strictly Constrained Generative Modeling via Split Augmented Langevin Sampling

2025-05-23

AI Summary

Enforcing exact physical conservation laws (e.g., energy, mass) remains challenging in scientific generative modeling, particularly within diffusion-based sampling frameworks. Method: This paper introduces Split Augmented Langevin (SAL), the first diffusion sampling framework that rigorously enforces hard physical constraints layer-by-layer during generation. SAL integrates variational Langevin dynamics with primal-dual optimization and variable splitting to construct a theoretically grounded, split-augmented dual sampling algorithm, providing provable convergence guarantees—departing from conventional soft-penalty approaches. Results: Experiments demonstrate that SAL significantly improves forecast accuracy in data assimilation for complex physical systems while strictly preserving conserved quantities. Moreover, it resolves feasibility issues in high-dimensional optimal control problems. By ensuring verifiable compliance with physical laws and enabling broad applicability across domains, SAL establishes a new, principled paradigm for physics-informed generative modeling.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Deep Generative Models & Autoencoders
• Natural Language Processing: Generation

Application Category

• Search and Retrieval-Augmented AI: Large language models for search
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

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.

Problem

• Enforcing physical constraints in generative models
• Guaranteeing plausibility of generated outputs
• Applying constraints to diffusion models effectively

Innovation

• Split Augmented Langevin sampling enforces constraints
• Primal-dual algorithm with convergence guarantees
• Constrained diffusion models conserve energy and mass