While the popularity of physics-informed neural networks (PINNs) is steadily rising, to this date, PINNs have not been successful in simulating multi-scale and singular perturbation problems. In this work, we present a new training paradigm referred to as "gradient boosting" (GB), which significantly enhances the performance of physics informed neural networks (PINNs). Rather than learning the solution of a given PDE using a single neural network directly, our algorithm employs a sequence of neural networks to achieve a superior outcome. This approach allows us to solve problems presenting great challenges for traditional PINNs. Our numerical experiments demonstrate the effectiveness of our algorithm through various benchmarks, including comparisons with finite element methods and PINNs. Furthermore, this work also unlocks the door to employing ensemble learning techniques in PINNs, providing opportunities for further improvement in solving PDEs.

翻译：尽管物理信息神经网络（PINNs）的普及度稳步上升，但迄今为止，PINNs在模拟多尺度和奇异摄动问题方面仍未取得成功。在这项工作中，我们提出了一种名为“梯度提升”（GB）的新型训练范式，显著提升了物理信息神经网络（PINNs）的性能。我们的算法并非直接使用单一神经网络学习给定PDE的解，而是采用一系列神经网络来实现更优的结果。这种方法使我们能够解决传统PINNs面临巨大挑战的问题。我们的数值实验通过多种基准测试（包括与有限元方法和PINNs的比较）证明了该算法的有效性。此外，这项工作也为在PINNs中采用集成学习技术打开了大门，为进一步改进PDE求解提供了机遇。