In this work we propose an extension of physics informed supervised learning strategies to parametric partial differential equations. Indeed, even if the latter are indisputably useful in many applications, they can be computationally expensive most of all in a real-time and many-query setting. Thus, our main goal is to provide a physics informed learning paradigm to simulate parametrized phenomena in a small amount of time. The physics information will be exploited in many ways, in the loss function (standard physics informed neural networks), as an augmented input (extra feature employment) and as a guideline to build an effective structure for the neural network (physics informed architecture). These three aspects, combined together, will lead to a faster training phase and to a more accurate parametric prediction. The methodology has been tested for several equations and also in an optimal control framework.

翻译：本文提出了一种将物理信息监督学习策略扩展到参数偏微分方程的方法。尽管这些方程在诸多应用中具有不可否认的实用性，但在实时和多查询场景下其计算成本往往高昂。因此，我们的主要目标是提供一种基于物理信息的学习范式，以在短时间内模拟参数化现象。物理信息将通过多种方式被利用：在损失函数中（标准物理信息神经网络）、作为增强输入（额外特征引入）以及作为构建神经网络有效结构的指导（物理信息架构）。这三个方面相结合，将加速训练阶段并提高参数预测的准确性。该方法已在多个方程以及最优控制框架下进行了测试。