We develop inductive biases for the machine learning of complex physical systems based on the port-Hamiltonian formalism. To satisfy by construction the principles of thermodynamics in the learned physics (conservation of energy, non-negative entropy production), we modify accordingly the port-Hamiltonian formalism so as to achieve a port-metriplectic one. We show that the constructed networks are able to learn the physics of complex systems by parts, thus alleviating the burden associated to the experimental characterization and posterior learning process of this kind of systems. Predictions can be done, however, at the scale of the complete system. Examples are shown on the performance of the proposed technique.

Summary: One of the main difficulties that 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 (PGD)] 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 proper orthogonal decomposition local approximations.The resulting method can be seen as a sort of a priori manifold learning or nonlinear dimensionality reduction technique. Examples are shown that demonstrate pros and cons of each strategy for different problems.

Learning and reasoning about physical phenomena is still a challenge in robotics development, and computational sciences play a capital role in the search for accurate methods able to provide explanations for past events and rigorous forecasts of future situations. We propose a thermodynamics-informed active learning strategy for fluid perception and reasoning from observations. As a model problem, we take the sloshing phenomena of different fluids contained in a glass. Starting from full-field and high-resolution synthetic data for a particular fluid, we develop a method for the tracking (perception) and simulation (reasoning) of any previously unseen liquid whose free surface is observed with a commodity camera. This approach demonstrates the importance of physics and knowledge not only in data-driven (gray-box) modeling but also in real-physics adaptation in low-data regimes and partial observations of the dynamics. The presented method is extensible to other domains such as the development of cognitive digital twins able to learn from observation of phenomena for which they have not been trained explicitly.

Physics perception very often faces the problem that only limited data or partial measurements on the scene are available. In this work, we propose a strategy to learn the full state of sloshing liquids from measurements of the free surface. Our approach is based on recurrent neural networks (RNN) that project the limited information available to a reduced-order manifold to not only reconstruct the unknown information but also be capable of performing fluid reasoning about future scenarios in real-time. To obtain physically consistent predictions, we train deep neural networks on the reduced-order manifold that, through the employ of inductive biases, ensure the fulfillment of the principles of thermodynamics. RNNs learn from history the required hidden information to correlate the limited information with the latent space where the simulation occurs. Finally, a decoder returns data to the high-dimensional manifold, to provide the user with insightful information in the form of augmented reality. This algorithm is connected to a computer vision system to test the performance of the proposed methodology with real information, resulting in a system capable of understanding and predicting future states of the observed fluid in real-time.