Abstract
Physically Guided Neural Networks with Internal Variables are SciML tools that use only observable data for training and and have the capacity to unravel internal state relations. They incorporate physical knowledge both by prescribing the model architecture and using loss regularization, thus endowing certain specific neurons with a physical meaning as internal state variables. Despite their potential, these models face challenges in scalability when applied to high-dimensional data such as fine-grid spatial fields or time-evolving systems. In this work, we propose some enhancements to the PGNNIV framework that address these scalability limitations through reduced-order modeling techniques. Specifically, we introduce alternatives to the original decoder structure using spectral decomposition, POD, and pretrained autoencoder-based mappings. These surrogate decoders offer varying trade-offs between computational efficiency, accuracy, noise tolerance, and generalization, while improving drastically the scalability. Additionally, we integrate model reuse via transfer learning and fine-tuning strategies to exploit previously acquired knowledge, supporting efficient adaptation to novel materials or configurations, and significantly reducing training time while maintaining or improving model performance. To illustrate these various techniques, we use a representative case governed by the nonlinear diffusion equation, using only observable data. Results demonstrate that the enhanced PGNNIV framework successfully identifies the underlying constitutive state equations while maintaining high predictive accuracy. It also improves robustness to noise, mitigates overfitting, and reduces computational demands. The proposed techniques can be tailored to various scenarios depending on data availability, resources, and specific modeling objectives, overcoming scalability challenges in all the scenarios.
Abstract (translated)
带有内部变量的物理引导神经网络（Physically Guided Neural Networks with Internal Variables，PGNNIV）是科学机器学习（SciML）工具的一种，它们仅使用可观察数据进行训练，并有能力揭示内部状态关系。这些模型通过规定模型架构和采用损失正则化来整合物理学知识，从而使某些特定的神经元具有作为内部状态变量的物理意义。尽管这类模型具备巨大潜力，但在应用于高维数据（如细网格空间场或时间演化系统）时面临可扩展性挑战。 在本工作中，我们提出了一些增强PGNNIV框架的方法以通过降阶建模技术解决这些可扩展性的限制。具体来说，我们引入了替代原始解码器结构的选项，包括光谱分解、POD（主成分分析）和基于预训练自动编码器映射的技术。这些替代解码器提供了计算效率、准确性、抗噪性以及泛化能力之间的不同权衡，并显著提高了可扩展性。 此外，我们通过迁移学习和微调策略整合了模型重用方法，利用先前获取的知识支持对新型材料或配置的高效适应，同时大幅减少训练时间并保持甚至提高模型性能。为了展示这些技术的应用效果，我们在一个由非线性扩散方程支配且仅使用可观察数据的情况下展示了代表性案例。 实验结果表明，增强后的PGNNIV框架能够成功识别潜在的状态方程式，并维持高预测准确性。它还提高了抗噪能力、缓解过拟合现象并减少计算需求。提出的这些技术可以根据数据可用性、资源和具体建模目标的不同情况进行调整，在所有情况下都能克服可扩展性的挑战。 通过这些方法，PGNNIV框架不仅能够处理更复杂的数据集和物理问题，而且能够在保持高精度的同时显著提高模型的效率和鲁棒性。这使得它成为科学研究中处理大规模数据和复杂系统的强大工具。
URL
https://arxiv.org/abs/2508.00959
