ChromOps.jl：用于色谱模型的高阶模拟与离散前向灵敏度分析

中文摘要 AI 辅助

机理色谱模型对工艺开发具有重要价值，但基于梯度的参数估计与优化会受限于计算成本及推导目标函数梯度的工作量。为解决该问题，本文提出一款完全可微的Julia语言色谱求解器ChromOps.jl，结合了高阶空间离散化与离散前向灵敏度分析（DFSA）。在含空间位阻质量作用动力学的6组分离子交换色谱问题上，对两种高阶空间离散化方法——有限和 summation-by-parts（FD-SBP）与间断Galerkin谱元法（DG-SEM）进行对比测试，二者在构造问题上均达到理论收敛速率；FD-SBP在正向（原）与对偶值模拟中展现出更优的精度-工作量性能，且实现方式概念上更简单。DFSA的计算时间满足t_DFSA ≈ 1.4(1+N_p)t_fwd，其中N_p为单个ChromOps.jl块中传播的参数数量，最多可达24个，即每增加1个参数约需1.4次正向求解。由于原问题与对偶值求解共享同一代码路径，DFSA可在无需手动推导链式法则代码的情况下，提供用户定义目标的精确梯度，从而实现便捷的基于梯度的参数估计与优化。

英文摘要

Mechanistic chromatography models are valuable for process development, but gradient-based parameter estimation and optimization can be hindered by computational cost and the effort of deriving objective-function gradients. To address this concern, a fully differentiable Julia chromatography solver, ChromOps.jl, is presented that combines high-order spatial discretization with discrete forward sensitivity analysis (DFSA). Two high-order spatial discretizations, finite difference summation-by-parts (FD-SBP) and the discontinuous Galerkin spectral element method (DG-SEM), are compared on a 6-component ion-exchange chromatography problem with steric mass action kinetics. Both attain their theoretical convergence rates on manufactured problems, while FD-SBP shows favourable work-precision performance for both forward (primal) and dual-valued simulations and is conceptually simpler to implement. DFSA scales as tDFSA approx 1.4(1+Np)tfwd for up to Np = 24 parameters propagated in a single ForwardDiff.jl chunk, i.e. about 1.4 forward solves per additional parameter. Because the primal and dual-valued solves share the same code path, DFSA provides accurate gradients of user-defined objectives without any hand-derived chain-rule code, enabling accessible gradient-based parameter estimation and optimization.

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25 pages, 10 figures, preprint