Strictly Constrained Generative Modeling via Split Augmented Langevin Sampling
Matthieu Blanke, Yongquan Qu, Sara Shamekh, Pierre Gentine
TL;DR
This paper introduces CASAL, a new sampling algorithm that enforces physical constraints in generative models, improving their reliability for scientific applications by combining Langevin dynamics with constraint satisfaction.
Contribution
The paper develops a novel primal-dual sampling framework, CASAL, that enforces constraints during Langevin-based sampling, with theoretical analysis and practical applications to physical systems.
Findings
Enforcing constraints improves forecast accuracy in physical systems.
CASAL effectively preserves conserved quantities in data assimilation.
The method applies to non-convex feasibility problems in optimal control.
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 mathematical constraints. Leveraging the variational formulation of Langevin dynamics and Lagrangian duality, we propose Constrained Alternated Split Augmented Langevin (CASAL), a novel primal-dual sampling algorithm that enforces constraints progressively through variable splitting. We analyze our algorithm in Wasserstein space and derive explicit mixing time rates. While the method is developed…
Peer Reviews
Decision·ICLR 2026 Poster
**S1.** The paper effectively highlights the central difficulty in constrained generation, balancing feasibility with sample diversity, and offers a thoughtful critique of simple projection-based solutions. **S2.** It presents a theoretically grounded primal–dual Langevin formulation that unifies ideas from constrained optimization and generative sampling in a principled way. **S3.** The exposition is clear and precise, with mathematical reasoning and well-structured derivations that make the
**W1.** The paper does not include enough baselines or adequately discuss prior work on constrained generation. Comparable methods such as PCFM [2], D-Flow [3], DiffusionPDE [4], and ECI [1] are only briefly mentioned or omitted. ECI, for instance, also enforces hard constraints, while PCFM demonstrates constrained sampling under nonconvex conditions: both directly relevant for benchmarking. **W2.** The empirical analysis is limited, lacking standard quantitative metrics such as MMSE, SMSE, or
1) A split augmented Langevin formulation that enforces per-sample constraints while retaining exploration; a practical combination of variable splitting and projection for sampling.\\ 2) Coherent variational/duality setup with clear algorithms; sensible baselines demonstrating feasibility and distributional fidelity. 3) Motivation is explicit; notation and procedures are readable; appendices provide proofs, variants, and implementation notes. 4) Training-free and model-agnostic; integrates with
1) Projection robustness: many constraints need iterative or approximate $P_C$. Quantify how projection error affects feasibility and sampling bias; include experiments sweeping inner-iteration counts or projection tolerances. 2) Hyperparameter tuning: performance depends on $\rho$ and dual step $\eta$. Propose adaptive schedules driven by observed coupling gaps $\|x_t-z_t\|$ or constraint residuals, and compare against fixed/annealed baselines. 3) Experimental breadth: add compact, classical PD
1. The SAL algorithm achieves strict feasibility without sacrificing exploration in constrained generation, addressing the weakness of previous projected or penalty-based approaches through a well-motivated formulation. 2. The paper provides a clear variational interpretation of constrained sampling and offers theoretical guarantees supporting the soundness and effectiveness of the proposed method. 3. The experiments cover a broad range of tasks and settings, demonstrating that SAL maintains phy
1. None of the three experiments report quantitative metrics or summary tables comparing different methods. 2. Lines 329–333 mention several baselines intended for comparison, yet the most relevant one, the Primal–Dual Langevin method, does not appear in the reported experimental results. 3. Lines 420–421 note that ADMM is a classical solver for obstacle-avoidance problems and highlight its limitations, but the experiments result does not provide a direct comparison between ADMM and the SAL, whi
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Taxonomy
TopicsBayesian Methods and Mixture Models · Generative Adversarial Networks and Image Synthesis · Algorithms and Data Compression
MethodsDiffusion