AI 中文总结
针对物理约束生成模型推理时投影计算昂贵的问题，提出利用块稀疏雅可比和KKT系统结构，结合GPU稀疏优化实现高效非线性约束投影，加速约束采样。
AI 中文摘要
生成模型已成为物理模拟的可扩展替代方案，但它们无法保证输出遵守控制底层物理的守恒定律、边界条件和非线性不变量。约束采样弥补了这一差距，在推理时精确执行此类约束而无需重新训练，但计算成本高昂：投影、校正和轨迹优化步骤在采样过程中重复进行，对于非线性约束，这些步骤变得昂贵。标准机器学习框架加剧了这一问题：其密集张量代数和有限的稀疏求解器组合性掩盖了物理约束自然诱导的结构，使得高效的批量非线性优化在实践中难以实现。我们通过利用样本批处理和局部PDE耦合在投影子问题中诱导的结构（即块稀疏雅可比和KKT系统）来解决这一瓶颈，使用this http URL暴露该结构，并通过this http URL和GPU稀疏分解求解所得的稀疏非线性规划。应用于物理约束流匹配（PCFM），在具有线性、非线性、一维和二维约束的PDE基准测试中，该方法在保持约束满足的同时加速了非线性约束投影。这些结果表明，稀疏GPU非线性优化是科学机器学习中约束生成采样的实用基础。
英文摘要
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.
