On the application of Physically-Guided Neural Networks with Internal Variables to Continuum Problems

Abstract

Predictive physics has been historically based upon the development of mathematical models that describe the evolution of a system under certain external stimuli and constraints. The structure of such mathematical models relies on a set of physical hypotheses that are assumed to be fulfilled by the system within a certain range of environmental conditions. A new perspective is now raising that uses physical knowledge to inform the data prediction capability of Machine Learning tools, coined as Scientific Machine Learning.

A particular approach in this context is the use of Physically-Guided Neural Networks with Internal Variables, where universal physical laws are used as constraints to a given neural network, in such a way that some neuron values can be interpreted as internal state variables of the system. This endows the network with unraveling capacity, as well as better predictive properties such as faster convergence, fewer data needs and additional noise filtering. Besides, only observable data are used to train the network, and the internal state equations may be extracted as a result of the training process, so there is no need to make explicit the particular structure of the internal state model, while getting solutions consistent with Physics.

We extend here this methodology to continuum physical problems driven by a general set of partial differential equations, showing again its predictive and explanatory capacities when only using measurable values in the training set. Moreover, we show that the mathematical operators developed for image analysis in deep learning approaches can be used in a natural way and extended to consider standard functional operators in continuum Physics, thus establishing a common framework for both.

The methodology presented demonstrates its ability to discover the internal constitutive state equation for some problems, including heterogeneous, anisotropic and nonlinear features, while maintaining its predictive ability for the whole dataset coverage, with the cost of a single evaluation. As a consequence, microstructural material properties can be inferred from macroscopic measurement coming from sensors without the need of specific homogeneous test plans neither specimen extraction.