In biomedical engineering, earthquake prediction, and underground energy harvesting, it is crucial to indirectly estimate the physical properties of porous media since the direct measurement of those are usually impractical/prohibitive. Here we apply the physics-informed neural networks to solve the inverse problem with regard to the nonlinear Biot's equations. Specifically, we consider batch training and explore the effect of different batch sizes. The results show that training with small batch sizes, i.e., a few examples per batch, provides better approximations (lower percentage error) of the physical parameters than using large batches or the full batch. The increased accuracy of the physical parameters, comes at the cost of longer training time. Specifically, we find the size should not be too small since a very small batch size requires a very long training time without a corresponding improvement in estimation accuracy. We find that a batch size of 8 or 32 is a good compromise, which is also robust to additive noise in the data. The learning rate also plays an important role and should be used as a hyperparameter.
UTOPYA: A Multimodal Deep Learning Framework for Physics-Informed Anomaly Detection and Time-Series Prediction
Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU)（化学工程系，挪威科学与技术大学）；Department of Computing, Imperial College London（计算系，帝国理工学院伦敦分校）
Curvature-Aware Optimization for High-Accuracy Physics-Informed Neural Networks
Department of Information Engineering and Computer Science, University of Trento（特伦托大学信息工程与计算机科学系）；Division of Applied Mathematics, Brown University（布朗大学应用数学部）；Center for Biomedical Engineering, Brown University（布朗大学生物医学工程中心）；Institute of Mathematics, TU Berlin（柏林工业大学数学研究所）；ETH Zurich（苏黎世联邦理工学院）
A Physics-Informed Neural Network to Model Port Channels
Instituto de Estudos Avançados – University of São Paulo（圣保罗大学高等研究院）；Escola Politécnica – University of Sao Paulo（圣保罗大学理工学院）；Center for Artificial Intelligence (C4AI) – University of Sao Paulo（圣保罗大学人工智能中心（C4AI））；Instituto Oceanográfico – University of Sao Paulo（圣保罗大学海洋研究所）
Variational inference in neural functional prior using normalizing flows: Application to differential equation and operator learning problems
基于物理学的深度学习最近已经成为利用观测数据和可用物理定律的有效工具。物理信息神经网络（PINN）和深度算子网络（DeepONets）是两个这样的模型。前者通过自动微分对物理定律进行编码，后者从数据中学习隐藏的物理规律。通常，噪声和有限的观测数据以及神经网络（NN）中的过度参数化会导致深度学习模型预测的不确定性。在[1]中，提出了一种基于{生成对抗网络}（GAN）的贝叶斯框架，作为量化PINN和DeepONets预测中的不确定性的统一模型。具体来说，[1]中提出的方法有两个阶段：（1）先验学习，和（2）后验估计。在第一阶段，使用GANs从规定的函数分布（例如，高斯过程，或从历史数据和可用的物理。在第二阶段，利用Hamilton Monte Carlo（HMC）方法估计GANs的潜在空间中的后验。然而，香草HMC不支持小批量训练，这限制了它在大数据问题中的应用。在目前的工作中，我们建议在变分推理的背景下使用归一化流（NF）模型，这自然能够实现小批量训练，作为HMC在GANs的潜在空间中进行后验估计的替代方案。一系列的数值实验，包括一个非线性微分方程问题和一个100维达西问题，进行了证明，NF与全/小批量训练能够达到类似的精度作为“黄金规则”的HMC。
