Substitution of well-grounded theoretical models by data-driven predictions is not as simple in engineering and sciences as it is in social and economic fields. Scientific problems suffer many times from paucity of data, while they may involve a large number of variables and parameters that interact in complex and non-stationary ways, obeying certain physical laws. Moreover, a physically-based model is not only useful for making predictions, but to gain knowledge by the interpretation of its structure, parameters, and mathematical properties. The solution to these shortcomings seems to be the seamless blending of the tremendous predictive power of the data-driven approach with the scientific consistency and interpretability of physically-based models.

We use here the concept of Physically-Guided Neural Networks (PGNN) to predict the input–output relation in a physical system, while, at the same time, fulfilling the physical constraints. With this goal, the internal hidden state variables of the system are associated with a set of internal neuron layers, whose values are constrained by known physical relations, as well as any additional knowledge on the system. Furthermore, when having enough data, it is possible to infer knowledge about the internal structure of the system and, if parameterized, to predict the state parameters for a particular input–output relation. We show that this approach, besides getting physically-based predictions, accelerates the training process, reduces the amount of data required to get similar accuracy, partly filters the intrinsic noise in the experimental data and improves its extrapolation capacity.

用数据驱动的预测取代有充分根据的理论模型在工程和科学领域并不像在社会和经济领域那样简单。缺乏数据使科学问题遭受了很多次困扰，而科学问题可能涉及大量变量和参数，这些变量和参数以复杂且不稳定的方式相互作用，遵循某些物理定律。此外，基于物理的模型不仅可用于进行预测，而且可通过解释其结构，参数和数学属性来获取知识。解决这些缺点的方法似乎是将数据驱动方法的巨大预测能力与基于物理的模型的科学一致性和可解释性无缝地融合在一起。

我们在这里使用物理引导神经网络（PGNN）的概念来预测物理系统中的输入输出关系，同时满足物理约束。为了这个目标，系统的内部隐藏状态变量与一组内部神经元层相关联，内部神经元层的值受已知的物理关系以及系统上的任何其他知识的约束。此外，当有足够的数据时，就可以推断出有关系统内部结构的知识，并且如果进行了参数化，则可以预测特定输入输出关系的状态参数。我们表明，这种方法除了获得基于身体的预测之外，还可以加快训练过程，减少获得相似精度所需的数据量，外推能力。