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.

翻译：物种传输模型通常将偏微分方程与受阻传输理论的关系相结合，以量化复杂纳米孔系统中的电迁移、对流和扩散输运；然而，这些公式往往是主导动力学的重大简化，导致基于PDE的模型泛化性能较差。鉴于物理科学领域对深度学习方法日益增长的兴趣，我们开发了一种基于机器学习的方法来表征跨纳米孔膜的离子传输。我们提出的框架以注意力增强神经微分方程为核心，通过引入基于电中性的归纳偏置，相较于传统的基于PDE的方法提高了泛化性能。此外，我们研究了注意力机制在揭示不同混合组分中具有物理意义的离子配对关系方面的作用。进一步地，我们探讨了基于PDE模型的模拟数据预训练的重要性，以及硬归纳偏置与软归纳偏置的性能优势。我们的结果表明，物理信息深度学习解决方案可以超越其经典的基于PDE的对应方法，并为跨多种应用场景的复杂传输现象建模提供了有前景的途径。