Traditional numerical schemes for simulating fluid flow and transport in
porous media can be computationally expensive. Advances in machine learning for
scientific computing have the potential to help speed up the simulation time in
many scientific and engineering fields. DeepONet has recently emerged as a
powerful tool for accelerating the solution of partial differential equations
(PDEs) by learning operators (mapping between function spaces) of PDEs. In this
work, we learn the mapping between the space of flux functions of the
Buckley-Leverett PDE and the space of solutions (saturations). We use
Physics-Informed DeepONets (PI-DeepONets) to achieve this mapping without any
paired input-output observations, except for a set of given initial or boundary
conditions; ergo, eliminating the expensive data generation process. By
leveraging the underlying physical laws via soft penalty constraints during
model training, in a manner similar to Physics-Informed Neural Networks
(PINNs), and a unique deep neural network architecture, the proposed
PI-DeepONet model can predict the solution accurately given any type of flux
function (concave, convex, or non-convex) while achieving up to four orders of
magnitude improvements in speed over traditional numerical solvers. Moreover,
the trained PI-DeepONet model demonstrates excellent generalization qualities,
rendering it a promising tool for accelerating the solution of transport
problems in porous media.

本文介绍了一种新的深度学习算法 ——PI-DeepONets 用于求解 Buckley-Leverett 偏微分方程并在多个科学和工程领域实现了四个数量级的速度提升，为传输问题研究提供了有前途的工具。

通过通量函数算符学习多孔介质中多相流传输的泛解决方案

Learning Generic Solutions for Multiphase Transport in Porous Media via  the Flux Functions Operator

Deep operator networks (DeepONets) are receiving increased attention thanks
to their demonstrated capability to approximate nonlinear operators between
infinite-dimensional Banach spaces. However, despite their remarkable early
promise, they typically require large training data-sets consisting of paired
input-output observations which may be expensive to obtain, while their
predictions may not be consistent with the underlying physical principles that
generated the observed data. In this work, we propose a novel model class
coined as physics-informed DeepONets, which introduces an effective
regularization mechanism for biasing the outputs of DeepOnet models towards
ensuring physical consistency. This is accomplished by leveraging automatic
differentiation to impose the underlying physical laws via soft penalty
constraints during model training. We demonstrate that this simple, yet
remarkably effective extension can not only yield a significant improvement in
the predictive accuracy of DeepOnets, but also greatly reduce the need for
large training data-sets. To this end, a remarkable observation is that
physics-informed DeepONets are capable of solving parametric partial
differential equations (PDEs) without any paired input-output observations,
except for a set of given initial or boundary conditions. We illustrate the
effectiveness of the proposed framework through a series of comprehensive
numerical studies across various types of PDEs. Strikingly, a trained physics
informed DeepOnet model can predict the solution of $\mathcal{O}(10^3)$
time-dependent PDEs in a fraction of a second -- up to three orders of
magnitude faster compared a conventional PDE solver. The data and code
accompanying this manuscript are publicly available at
https://github.com/PredictiveIntelligenceLab/Physics-informed-DeepONets.

本文提出了一种被称为物理学知识不同 DeepONets 的新模型类，通过使用自动差分在模型训练期间施加软惩罚约束来实现重力定律，其将 DeepONet 模型输出偏向于确保物理一致性，进而显著提高 DeepONets 的预测准确性，并大大减少了大型训练数据集的需求。