We revisit the original approach of using deep learning and neural networks to solve differential equations by incorporating the knowledge of the equation. This is done by adding a dedicated term to the loss function during the optimization procedure in the training process. The so-called physics-informed neural networks (PINNs) are tested on a variety of academic ordinary differential equations in order to highlight the benefits and drawbacks of this approach with respect to standard integration methods. We focus on the possibility to use the least possible amount of data into the training process. The principles of PINNs for solving differential equations by enforcing physical laws via penalizing terms are reviewed. A tutorial on a simple equation model illustrates how to put into practice the method for ordinary differential equations. Benchmark tests show that a very small amount of training data is sufficient to predict the solution when the non linearity of the problem is weak. However, this is not the case in strongly non linear problems where a priori knowledge of training data over some partial or the whole time integration interval is necessary.

翻译：我们重新审视了利用深度学习与神经网络求解微分方程的原始方法，该方法通过将方程知识融入优化过程，在训练阶段向损失函数添加专用项来实现。将所谓的物理信息神经网络应用于多种学术常微分方程，以突出该方法相较于标准积分法的优势与不足。我们重点关注在训练过程中尽可能减少数据使用的可能性。回顾了通过惩罚项强制执行物理定律的PINNs求解微分方程原理。通过一个简单方程模型的教程，阐明了如何将该方法应用于常微分方程实践。基准测试表明，当问题非线性较弱时，极少量训练数据即可预测解。然而，在强非线性问题中则不然，此时需要预先获取部分或整个时间积分区间上的训练数据知识。