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This research tackles the computational bottleneck in physics-constrained generative models by leveraging block-sparse Jacobian structures to accelerate nonlinear projection steps during inference. By integrating specialized sparse solvers with GPU factorization, the method maintains strict adherence to physical laws like conservation and boundary conditions without requiring model retraining. The work establishes a practical foundation for scaling scientific machine learning applications where efficiency and physical fidelity are critical.
SNAP-FM: Sparse Nonlinear Accelerated Projection for Physics-Constrained Generative Modeling
arXiv:2607.00095v2 Announce Type: replace-cross 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 ExaModels.jl and solving the resulting sparse nonlinear programs with MadNLP.jl 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.
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.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Computational Engineering, Finance, and Science (cs.CE)
arXiv:2607.00095 [cs.LG]
(or arXiv:2607.00095v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.00095 Focus to learn more arXiv-issued DOI via DataCite
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