We develop a method to learn physical systems from data that employs feedforward neural networks and whose predictions comply with the first and second principles of thermodynamics. The method employs a minimum amount of data by enforcing the metriplectic structure of dissipative Hamiltonian systems in the form of the socalled General Equation for the Non-Equilibrium Reversible-Irreversible Coupling, GENERIC (Öttinger and Grmela (1997) [36]). The method does not need to enforce any kind of balance equation, and thus no previous knowledge on the nature of the system is needed. Conservation of energy and dissipation of entropy in the prediction of previously unseen situations arise as a natural by-product of the structure of the method. Examples of the performance of the method are shown that comprise conservative as well as dissipative systems, discrete as well as continuous ones.
We present an algorithm to learn the relevant latent variables of a large-scale discretized physical system and predict its time evolution using thermodynamically-consistent deep neural networks. Our method relies on sparse autoencoders, which reduce the dimensionality of the full order model to a set of sparse latent variables with no prior knowledge of the coded space dimensionality. Then, a second neural network is trained to learn the metriplectic structure of those reduced physical variables and predict its time evolution with a so-called structure-preserving neural network. This data-based integrator is guaranteed to conserve the total energy of the system and the entropy inequality, and can be applied to both conservative and dissipative systems.The integrated paths can then be decoded to the original full-dimensional manifold and be compared to the ground truth solution. This method is tested with two examples applied to fluid and solid mechanics. c
One of the main difficulties a reduced order method could face is the poor separability of the solution. This problem is common to both a posteriori model order reduction (Proper Orthogonal Decomposition, Reduced Basis) and a priori (Proper Generalized Decomposition) model order reduction. Early approaches to solve it include the construction of local reduced order models in the framework of POD. We present here an extension of local models in a PGD -and thus, a priori-context. Three different strategies are introduced to estimate the size of the different patches or regions in the solution manifold where PGD is applied. As will be noticed, no gluing or special technique is needed to deal with the resulting set of local reduced order models, in contrast to most POD local approximations. The resulting method can be seen as a sort of a priori manifold learning or non-linear dimensionality reduction technique. Examples are shown that demonstrate pros and cons of each strategy for different problems.KEY WORDS: local model order reduction, proper generalized decomposition, kernel principal component analysis, non linear dimensionality reduction.
Digital twins can be defined as digital representations of physical entities that employ real-time data to enable understanding of the operating conditions of these entities.Here we present a particular type of digital twin that involves a combination of computer vision, scientific machine learning and augmented reality. This novel digital twin is able, therefore, to see, to interpret what it sees-and, if necessary, to correct the model it is equipped with-and presents the resulting information in the form of augmented reality.The computer vision capabilities allow the twin to receive data continuously. As any other digital twin, it is equipped with one or more models so as to assimilate data. However, if persistent deviations from the predicted values are found, the proposed methodology is able to correct on the fly the existing models, so as to accommodate them to the measured reality.Finally, the suggested methodology is completed with augmented reality capabilities so as to render a completely new type of digital twin. These concepts are tested against a proof-of-concept model consisting on a non-linear, hyperelastic beam subjected to moving loads whose exact position is to be determined.
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