Species transport models typically combine partial differential equations
(PDEs) with relations from hindered transport theory to quantify
electromigrative, convective, and diffusive transport through complex
nanoporous systems; however, these formulations are frequently substantial
simplifications of the governing dynamics, leading to the poor generalization
performance of PDE-based models. Given the growing interest in deep learning
methods for the physical sciences, we develop a machine learning-based approach
to characterize ion transport across nanoporous membranes. Our proposed
framework centers around attention-enhanced neural differential equations that
incorporate electroneutrality-based inductive biases to improve generalization
performance relative to conventional PDE-based methods. In addition, we study
the role of the attention mechanism in illuminating physically-meaningful
ion-pairing relationships across diverse mixture compositions. Further, we
investigate the importance of pre-training on simulated data from PDE-based
models, as well as the performance benefits from hard vs. soft inductive
biases. Our results indicate that physics-informed deep learning solutions can
outperform their classical PDE-based counterparts and provide promising avenues
for modelling complex transport phenomena across diverse applications.

通过基于深度学习的注意力增强神经微分方程模型，我们研究离子在纳米孔膜中的运输，并探讨了注意力机制在揭示不同混合物组成下的离子配对关系中的作用。结果表明，与传统的基于偏微分方程的模型相比，基于物理学原理的深度学习解决方案在建模复杂运输现象方面表现更好，并为各种应用提供了有前途的途径。