SNAP-FM: Sparse Nonlinear Accelerated Projection for Physics-Constrained Generative Modeling
Generative models used as surrogates for physical simulation often fail to enforce physics constraints. Constrained sampling can enforce constraints at inference but is computationally expensive. This paper introduces SNAP-FM, which leverages sparse GPU nonlinear optimization to accelerate constraint projection. Using block-sparse Jacobian and KKT systems with ExaModels.jl and MadNLP.jl, the method achieves faster nonlinear constraint projection on PDE benchmarks while maintaining accuracy.
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[Submitted on 30 Jun 2026]
Title:SNAP-FM: Sparse Nonlinear Accelerated Projection for Physics-Constrained Generative Modeling
View a PDF of the paper titled SNAP-FM: Sparse Nonlinear Accelerated Projection for Physics-Constrained Generative Modeling, by Alaina Kolli and 6 other authors
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Abstract:Generative models have emerged as scalable surrogates for physical simulation, yet they offer no guarantee that their outputs respect the conservation laws, boundary conditions, and nonlinear invariants that govern the underlying physics. Constrained sampling closes this gap, enforcing such constraints exactly at inference time without retraining, but at a computational cost: projection, correction, and trajectory-optimization steps are repeated during sampling, with these steps becoming expensive for nonlinear constraints. Standard ML frameworks exacerbate this: their dense tensor algebra and limited sparse solver composability obscure the structure that physical constraints naturally induce, making efficient batched nonlinear optimization difficult to realize in practice. We address this bottleneck by exploiting the structure that sample-wise batching and local PDE couplings induce in the projection subproblems -- namely, block-sparse Jacobian and KKT systems -- exposing this structure using this http URL and solving the resulting sparse nonlinear programs with this http URL and GPU sparse factorization. Applied to Physics-Constrained Flow Matching (PCFM), on PDE benchmarks with linear, nonlinear, one-dimensional, and two-dimensional constraints, this approach accelerates nonlinear constraint projection while maintaining constraint satisfaction. These results show that sparse GPU nonlinear optimization is a practical foundation for constrained generative sampling in scientific machine learning.
Subjects:
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Computational Engineering, Finance, and Science (cs.CE)
Cite as: arXiv:2607.00095 [cs.LG]
(or arXiv:2607.00095v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.00095
arXiv-issued DOI via DataCite (pending registration)
Submission history
From: Utkarsh Utkarsh [view email] [v1] Tue, 30 Jun 2026 19:35:09 UTC (490 KB)
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