Predicting and explaining nonlinear material response using deep Physically Guided Neural Networks with Internal Variables

Nonlinear materials are often difficult to model with classical state model theory because they have a complex and sometimes inaccurate physical and mathematical description or we simply do not know how to describe such materials in terms of relations between external and internal variables. In many disciplines, Neural Network methods have arisen as powerful tools to identify very complex and non-linear correlations. In this work, we use the very recently developed concept of Physically Guided Neural Networks with Internal Variables (PGNNIV) to discover constitutive laws using a model-free approach and training solely with measured force-displacement data. PGNNIVs make a particular use of the physics of the problem to enforce constraints on specific hidden layers and are able to make predictions without internal variable data. We demonstrate that PGNNIVs are capable of predicting both internal and external variables under unseen load scenarios, regardless of the nature of the material considered (linear, with hardening or softening behavior and hyperelastic), unravelling the constitutive law of the material hence explaining its nature altogether, placing the method in what is known as eXplainable Artificial Intelligence (XAI).

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

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 hysical 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 artificial neural networks. A particular extension of this data-driven approach is Physically-Guided Neural Networks with Internal Variables (PGNNIV): universal physical laws are used as constraints in the 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 processes, so there is no need to make explicit the particular structure of the internal state model. We extend this new methodology to continuum physical problems, showing again its predictive and explanatory capacities when only using measurable values in the training set. We show that the mathematical operators developed for image analysis in deep learning approaches can be used 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 and nonlinear features, while maintaining its predictive ability for the whole dataset coverage, with the cost of a single evaluation.

Identification of state functions by physically-guided neural networks with physically-meaningful internal layers

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 most 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-constrained neural networks (PCNN) 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, filters partly the intrinsic noise in the experimental data and provides improved extrapolation capacity.