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 \url{https://github.com/PredictiveIntelligenceLab/Physics-informed-DeepONets}.
翻译:深操作器网络 (DeepONets) 正在受到更多关注, 原因是它们展示了在无限的Banach空间之间接近非线性操作器的能力。 然而,尽管它们早期的预言令人瞩目的是, 它们通常需要大型的培训数据集, 包括配对输入输出观测, 获取成本可能很高, 而它们的预测可能与生成观测数据的基本物理原则不符。 在这项工作中, 我们提议了一个新型模型类, 以物理知情的 DeepONets 创建, 引入一个有效的正规化机制, 偏向 DeepOint模型的输出, 以确保物理一致性。 实现这个机制的办法是在模型培训期间, 利用自动区分, 通过软性刑罚限制强制实施基本物理法。 我们证明, 这个简单而显著有效的扩展不仅能显著提高DeepOintiots的预测准确性, 而且还大大降低了对大量培训数据数据集的需求。 为此, 我们提出一个引人注目的观察是, 物理知情的深度的深度的二等分数部分方程式(PDE) (P- II) 的第二分解度观测, 除了给定的一套或深底底底底底线 。