Francisco Chinesta

Sebastian Rodriguez, Borja Ferrandiz, Francisco Chinesta et al.

Structural Health Monitoring (SHM) aims at the real-time monitoring of the integrity of engineering structures, with Guided-waves (GWs) providing high sensitivity to damage presence and to ageing effects for thin-walled components. In conventional GW-based SHM, a bonded piezoelectric transducer (PZT) emits a short tone burst that produces an Initial Wave Packet (IWP) propagating through the structure. As this packet interacts with boundaries and potential damages, additional scattered wave packets are produced. A major limitation of such approaches lies in the simultaneous excitation of multiple dispersive GW modes by a single PZT, which significantly complicates signal interpretation and damage monitoring. In this context, this work proposes the Physics-Informed Single Atom Matching Pursuit (PISAMP) method, a signal decomposition method grounded in the physical principles governing wave propagation. In contrast with purely data-driven or numerically intensive techniques, the proposed approach embeds strong physical constraints into a low-dimensional and computationally efficient signal representation. This formulation enables the direct identification of key physically meaningful features, including modal wavenumber functions and propagation distances between actuator, damage and sensors. These extracted features, especially source-damage-sensor distances, allows to subsequently perform damage location using well established Elliptical Localization techniques. The principal novelty of this study lies in integrating wave propagation physics into a compact signal decomposition framework and developing an interpretable damage localization methodology for GW-SHM applications.

Quercus Hernández, Alberto Badías, Francisco Chinesta et al.

In this paper we present a deep learning method to predict the temporal evolution of dissipative dynamic systems. We propose using both geometric and thermodynamic inductive biases to improve accuracy and generalization of the resulting integration scheme. The first is achieved with Graph Neural Networks, which induces a non-Euclidean geometrical prior with permutation invariant node and edge update functions. The second bias is forced by learning the GENERIC structure of the problem, an extension of the Hamiltonian formalism, to model more general non-conservative dynamics. Several examples are provided in both Eulerian and Lagrangian description in the context of fluid and solid mechanics respectively, achieving relative mean errors of less than 3% in all the tested examples. Two ablation studies are provided based on recent works in both physics-informed and geometric deep learning.

Elias Cueto, Francisco Chinesta

Thermodynamics could be seen as an expression of physics at a high epistemic level. As such, its potential as an inductive bias to help machine learning procedures attain accurate and credible predictions has been recently realized in many fields. We review how thermodynamics provides helpful insights in the learning process. At the same time, we study the influence of aspects such as the scale at which a given phenomenon is to be described, the choice of relevant variables for this description or the different techniques available for the learning process.

Quercus Hernández, Alberto Badías, Francisco Chinesta et al.

The imminent impact of immersive technologies in society urges for active research in real-time and interactive physics simulation for virtual worlds to be realistic. In this context, realistic means to be compliant to the laws of physics. In this paper we present a method for computing the dynamic response of (possibly non-linear and dissipative) deformable objects induced by real-time user interactions in mixed reality using deep learning. The graph-based architecture of the method ensures the thermodynamic consistency of the predictions, whereas the visualization pipeline allows a natural and realistic user experience. Two examples of virtual solids interacting with virtual or physical solids in mixed reality scenarios are provided to prove the performance of the method.

Beatriz Moya, Alberto Badias, David Gonzalez et al.

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 analysis (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 (grey box) modeling but also in the correction for real physics adaptation in low data regimes and partial observations of the dynamics. The method presented 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.

Quercus Hernández, Alberto Badías, Francisco Chinesta et al.

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.

Jad Mounayer, Sebastian Rodriguez, Jerome Tomezyk et al.

Most existing latent-space models for dynamical systems require fixing the latent dimension in advance, they rely on complex loss balancing to approximate linear dynamics, and they don't regularize the latent variables. We introduce RRAEDy, a model that removes these limitations by discovering the appropriate latent dimension, while enforcing both regularized and linearized dynamics in the latent space. Built upon Rank-Reduction Autoencoders (RRAEs), RRAEDy automatically rank and prune latent variables through their singular values while learning a latent Dynamic Mode Decomposition (DMD) operator that governs their temporal progression. This structure-free yet linearly constrained formulation enables the model to learn stable and low-dimensional dynamics without auxiliary losses or manual tuning. We provide theoretical analysis demonstrating the stability of the learned operator and showcase the generality of our model by proposing an extension that handles parametric ODEs. Experiments on canonical benchmarks, including the Van der Pol oscillator, Burgers' equation, 2D Navier-Stokes, and Rotating Gaussians, show that RRAEDy achieves accurate and robust predictions. Our code is open-source and available at https://github.com/JadM133/RRAEDy. We also provide a video summarizing the main results at https://youtu.be/ox70mSSMGrM.

Hooman Danesh, Daniele Di Lorenzo, Francisco Chinesta et al.

Auxetic structures, known for their negative Poisson's ratio, exhibit effective elastic properties heavily influenced by their underlying structural geometry and base material properties. While periodic homogenization of auxetic unit cells can be used to investigate these properties, it is computationally expensive and limits design space exploration and inverse analysis. In this paper, surrogate models are developed for the real-time prediction of the effective elastic properties of auxetic unit cells with orthogonal voids of different shapes. The unit cells feature orthogonal voids in four distinct shapes, including rectangular, diamond, oval, and peanut-shaped voids, each characterized by specific void diameters. The generated surrogate models accept geometric parameters and the elastic properties of the base material as inputs to predict the effective elastic constants in real-time. This rapid evaluation enables a practical inverse analysis framework for obtaining the optimal design parameters that yield the desired effective response. The fast Fourier transform (FFT)-based homogenization approach is adopted to efficiently generate data for developing the surrogate models, bypassing concerns about periodic mesh generation and boundary conditions typically associated with the finite element method (FEM). The performance of the generated surrogate models is rigorously examined through a train/test split methodology, a parametric study, and an inverse problem. Finally, a graphical user interface (GUI) is developed, offering real-time prediction of the effective tangent stiffness and performing inverse analysis to determine optimal geometric parameters.

Sebastian Rodriguez, Mikhael Tannous, Jad Mounayer et al.

Unidirectional tapes surface roughness determines the evolution of the degree of intimate contact required for ensuring the thermoplastic molecular diffusion and the associated inter-tapes consolidation during manufacturing of composite structures. However, usual characterization of rough surfaces relies on statistical descriptors that even if they are able to represent the surface topology, they are not necessarily connected with the physics occurring at the interface during inter-tape consolidation. Thus, a key research question could be formulated as follows: Which roughness descriptors simultaneously enable tape classification-crucial for process control-and consolidation modeling via the inference of the evolution of the degree of intimate contact, itself governed by the process parameters?. For providing a valuable response, we propose a novel strategy based on the use of Rank Reduction Autoencoders (RRAEs), autoencoders with a linear latent vector space enforced by applying a truncated Singular Value Decomposition (SVD) to the latent matrix during the encoder-decoder training. In this work, we extract useful roughness descriptors by enforcing the latent SVD modes to (i) accurately represent the roughness after decoding, and (ii) allow the extraction of existing a priori knowledge such as classification or modelling properties.

Jad Mounayer, Alicia Tierz, Jerome Tomezyk et al.

Deterministic Rank Reduction Autoencoders (RRAEs) enforce by construction a regularization on the latent space by applying a truncated SVD. While this regularization makes Autoencoders more powerful, using them for generative purposes is counter-intuitive due to their deterministic nature. On the other hand, Variational Autoencoders (VAEs) are well known for their generative abilities by learning a probabilistic latent space. In this paper, we present Variational Rank Reduction Autoencoders (VRRAEs), a model that leverages the advantages of both RRAEs and VAEs. Our claims and results show that when carefully sampling the latent space of RRAEs and further regularizing with the Kullback-Leibler (KL) divergence (similarly to VAEs), VRRAEs outperform RRAEs and VAEs. Additionally, we show that the regularization induced by the SVD not only makes VRRAEs better generators than VAEs, but also reduces the possibility of posterior collapse. Our results include a synthetic dataset of a small size that showcases the robustness of VRRAEs against collapse, and three real-world datasets; the MNIST, CelebA, and CIFAR-10, over which VRRAEs are shown to outperform both VAEs and RRAEs on many random generation and interpolation tasks based on the FID score. We developed an open-source implementation of VRRAEs in JAX (Equinox), available at https://github.com/JadM133/RRAEs.git.

Di Jiang, Sebastian Rodriguez, Herve Colin et al.

Automotive engineering makes extensive use of numerical simulation throughout the design process. The development of numerical models, their validation against experimental tests, and their updating during vehicle and engine projects constitute a core engineering activity. However, this activity must continuously evolve to reduce costs and lead times. In this context, we propose a method for detecting faulty modules within a system-level simulation workflow, represented as a graph of 0D models, following model updates. The proposed method requires a very limited number of system simulations and can therefore be easily integrated into existing engineering processes. It is designed as a toolbox based on well established and widely validated techniques, including Dynamic Mode Decomposition commonly used for 3D model reduction, linear programming, and autoencoders.

Mohammed El Fallaki Idrissi, Jad Mounayer, Sebastian Rodriguez et al.

This paper presents a novel paradigm in simulation-based engineering sciences by introducing a new framework called Generative Parametric Design (GPD). The GPD framework enables the generation of new designs along with their corresponding parametric solutions given as a reduced basis. To achieve this, two Rank Reduction Autoencoders (RRAEs) are employed, one for encoding and generating the design or geometry, and the other for encoding the sparse Proper Generalized Decomposition (sPGD) mode solutions. These models are linked in the latent space using regression techniques, allowing efficient transitions between design and their associated sPGD modes. By empowering design exploration and optimization, this framework also advances digital and hybrid twin development, enhancing predictive modeling and real-time decision-making in engineering applications. The developed framework is demonstrated on two-phase microstructures, in which the multiparametric solutions account for variations in two key material parameters.

Edgar Chaillou, Sebastian Rodriguez, Yves Tourbier et al.

We present CRADIPOR, a numerical dispersion prediction tool for automotive crash simulations. Finite Element (FE) crash models are widely used throughout vehicle development, but their predictions are not strictly repeatable because of parallel computation and model complexity. As a result, performance criteria evaluated during post-processing may exhibit significant numerical dispersion, which complicates engineering decision-making. Although dispersion can be estimated by repeating the same simulation, this approach is generally impractical because of its high computational cost. This work therefore investigates a prediction tool that can be applied during routine crash-simulation post-processing without repeating the computation. The proposed approach relies on a Rank Reduction Autoencoder (RRAE) combined with supervised classification in order to identify regions sensitive to numerical dispersion. The comparative analysis suggests that the RRAE-based framework is more effective than the Random Forest baseline on the studied dataset. Among the tested signal representations, wavelet-based and slope-based inputs appear to be the most promising, with slope variations providing the best classification performance. These results support the use of structured latent representations for improving numerical-dispersion detection in automotive crash post-processing.

Francisco Chinesta, Antonio Falco, Daniela Falco-Pomares

We study numerical integration on $[0,1]$ by quantum amplitude estimation (QAE), focusing on the cost of constructing the amplitude oracle. Although QAE improves the statistical component of the integration error, this advantage is relevant only when the integrand has low encoding complexity. We introduce a hierarchy of grid function classes $\mathcal{G}_n^{(d)}$, defined by requiring the angle map $Θ_g:\{0,1\}^n\to[0,π]$ to be multilinear of degree at most $d$. Membership is classically checkable in $O(n2^n)$ time by the Walsh--Hadamard transform. For $g\in\mathcal{G}_n^{(d)}$, the encoding operator factorises into $\sum_{k=0}^d\binom{n}{k}$ multi-controlled $R_Y$ gates, interpolating between an affine $O(n)$ regime and the generic exponential regime. Combining this structure with classical discretisation estimates for $g\in C^α[0,1]$, we obtain a depth-versus-accuracy trade-off: gate count $O((\log(1/\varepsilon))^d\varepsilon^{-1})$ suffices to achieve $\varepsilon$-accuracy with constant probability. For $d=1$ this becomes $O(\varepsilon^{-1}\log(1/\varepsilon))$, improving over classical Monte Carlo for every $α\ge1$. We also prove an unconditional separation: $\mathcal{G}_n^{(1)}$ contains functions of Sobolev regularity $s<1/2$ for which the quantum oracle cost is $O(1/\varepsilon)$, whereas classical deterministic or randomised quadrature requires $Ω(\varepsilon^{-1/s})$ evaluations. These results identify explicit integrand classes for which the full cost of QAE-based integration, including state preparation, is asymptotically better than classical methods. Experiments on SpinQ Triangulum and IBM Kingston illustrate the hierarchy at $n=2$: circuits inside $\mathcal{G}_n^{(d)}$ run successfully, while those exceeding the Triangulum coherence budget fail as predicted.

Carlos Bermejo-Barbanoj, Beatriz Moya, Alberto Badías et al.

We present a method to increase the resolution of measurements of a physical system and subsequently predict its time evolution using thermodynamics-aware neural networks. Our method uses adversarial autoencoders, which reduce the dimensionality of the full order model to a set of latent variables that are enforced to match a prior, for example a normal distribution. Adversarial autoencoders are seen as generative models, and they can be trained to generate high-resolution samples from low-resoution inputs, meaning they can address the so-called super-resolution problem. Then, a second neural network is trained to learn the physical structure of the latent variables and predict their temporal evolution. This neural network is known as an structure-preserving neural network. It learns the metriplectic-structure of the system and applies a physical bias to ensure that the first and second principles of thermodynamics are fulfilled. The integrated trajectories are decoded to their original dimensionality, as well as to the higher dimensionality space produced by the adversarial autoencoder and they are compared to the ground truth solution. The method is tested with two examples of flow over a cylinder, where the fluid properties are varied between both examples.

Pau Urdeitx, Icíar Alfaro, David González et al.

The development of inductive biases has been shown to be a very effective way to increase the accuracy and robustness of neural networks, particularly when they are used to predict physical phenomena. These biases significantly increase the certainty of predictions, decrease the error made and allow considerably smaller datasets to be used. There are a multitude of methods in the literature to develop these biases. One of the most effective ways, when dealing with physical phenomena, is to introduce physical principles of recognised validity into the network architecture. The problem becomes more complex without knowledge of the physical principles governing the phenomena under study. A very interesting possibility then is to turn to the principles of thermodynamics, which are universally valid, regardless of the level of abstraction of the description sought for the phenomenon under study. To ensure compliance with the principles of thermodynamics, there are formulations that have a long tradition in many branches of science. In the field of rheology, for example, two main types of formalisms are used to ensure compliance with these principles: one-generator and two-generator formalisms. In this paper we study the advantages and disadvantages of each, using classical problems with known solutions and synthetic data.

Jad Mounayer, Sebastian Rodriguez, Chady Ghnatios et al.

The choice of an appropriate bottleneck dimension and the application of effective regularization are both essential for Autoencoders to learn meaningful representations from unlabeled data. In this paper, we introduce a new class of deterministic autoencoders, Rank Reduction Autoencoders (RRAEs), which regularize their latent spaces by employing a truncated singular value decomposition (SVD) during training. In RRAEs, the bottleneck is defined by the rank of the latent matrix, thereby alleviating the dependence of the encoder/decoder architecture on the bottleneck size. This approach enabled us to propose an adaptive algorithm (aRRAEs) that efficiently determines the optimal bottleneck size during training. We empirically demonstrate that both RRAEs and aRRAEs are stable, scalable, and reliable, as they do not introduce any additional training hyperparameters. We evaluate our proposed architecture on a synthetic data set, as well as on MNIST, Fashion MNIST, and CelebA. Our results show that RRAEs offer several advantages over Vanilla AEs with both large and small latent spaces, and outperform other regularizing AE architectures.

Guillaume de Romémont, Florent Renac, Jorge Nunez et al.

This paper presents a data-driven finite volume method for solving 1D and 2D hyperbolic partial differential equations. This work builds upon the prior research incorporating a data-driven finite-difference approximation of smooth solutions of scalar conservation laws, where optimal coefficients of neural networks approximating space derivatives are learned based on accurate, but cumbersome solutions to these equations. We extend this approach to flux-limited finite volume schemes for hyperbolic scalar and systems of conservation laws. We also train the discretization to efficiently capture discontinuous solutions with shock and contact waves, as well as to the application of boundary conditions. The learning procedure of the data-driven model is extended through the definition of a new loss, paddings and adequate database. These new ingredients guarantee computational stability, preserve the accuracy of fine-grid solutions, and enhance overall performance. Numerical experiments using test cases from the literature in both one- and two-dimensional spaces demonstrate that the learned model accurately reproduces fine-grid results on very coarse meshes.

Elias Al Ghazal, Jad Mounayer, Beatriz Moya et al.

Modeling and predicting the dynamics of complex multiscale systems remains a significant challenge due to their inherent nonlinearities and sensitivity to initial conditions, as well as limitations of traditional machine learning methods that fail to capture high frequency behaviours. To overcome these difficulties, we propose three approaches for multiscale learning. The first leverages the Partition of Unity (PU) method, integrated with neural networks, to decompose the dynamics into local components and directly predict both macro- and micro-scale behaviors. The second applies the Singular Value Decomposition (SVD) to extract dominant modes that explicitly separate macro- and micro-scale dynamics. Since full access to the data matrix is rarely available in practice, we further employ a Sparse High-Order SVD to reconstruct multiscale dynamics from limited measurements. Together, these approaches ensure that both coarse and fine dynamics are accurately captured, making the framework effective for real-world applications involving complex, multi-scale phenomena and adaptable to higher-dimensional systems with incomplete observations, by providing an approximation and interpretation in all time scales present in the phenomena under study.

Alicia Tierz, Jad Mounayer, Beatriz Moya et al.

Generative thermal design for complex geometries is fundamental in many areas of engineering, yet it faces two main challenges: the high computational cost of high-fidelity simulations and the limitations of conventional generative models. Approaches such as autoencoders (AEs) and variational autoencoders (VAEs) often produce unstructured latent spaces with discontinuities, which restricts their capacity to explore designs and generate physically consistent solutions. To address these limitations, we propose a hybrid framework that combines Variational Rank-Reduction Autoencoders (VRRAEs) with Deep Operator Networks (DeepONets). The VRRAE introduces a truncated SVD within the latent space, leading to continuous, interpretable, and well-structured representations that mitigate posterior collapse and improve geometric reconstruction. The DeepONet then exploits this compact latent encoding in its branch network, together with spatial coordinates in the trunk network, to predict temperature gradients efficiently and accurately. This hybrid approach not only enhances the quality of generated geometries and the accuracy of gradient prediction, but also provides a substantial advantage in inference efficiency compared to traditional numerical solvers. Overall, the study underscores the importance of structured latent representations for operator learning and highlights the potential of combining generative models and operator networks in thermal design and broader engineering applications.

Alicia Tierz, Iciar Alfaro, David González et al.

Thermodynamics-informed neural networks employ inductive biases for the enforcement of the first and second principles of thermodynamics. To construct these biases, a metriplectic evolution of the system is assumed. This provides excellent results, when compared to uninformed, black box networks. While the degree of accuracy can be increased in one or two orders of magnitude, in the case of graph networks, this requires assembling global Poisson and dissipation matrices, which breaks the local structure of such networks. In order to avoid this drawback, a local version of the metriplectic biases has been developed in this work, which avoids the aforementioned matrix assembly, thus preserving the node-by-node structure of the graph networks. We apply this framework for examples in the fields of solid and fluid mechanics. Our approach demonstrates significant computational efficiency and strong generalization capabilities, accurately making inferences on examples significantly different from those encountered during training.

Tarek Frahi, Abel Sancarlos, Matthieu Galle et al.

The present paper aims at analyzing the topological content of the complex trajectories that weeder-autonomous robots follow in operation. We will prove that the topological descriptors of these trajectories are affected by the robot environment as well as by the robot state, with respect to maintenance operations. Topological Data Analysis will be used for extracting the trajectory descriptors, based on homology persistence. Then, appropriate metrics will be applied in order to compare that topological representation of the trajectories, for classifying them or for making efficient pattern recognition.

Beatriz Moya, Alberto Badias, David Gonzalez et al.

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 so as to not only reconstruct the unknown information, but also to 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 back to the high-dimensional manifold, so as 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.

Abel Sancarlos, Morgan Cameron, Jean-Marc Le Peuvedic et al.

The concept of Hybrid Twin (HT) has recently received a growing interest thanks to the availability of powerful machine learning techniques. This twin concept combines physics-based models within a model-order reduction framework-to obtain real-time feedback rates-and data science. Thus, the main idea of the HT is to develop on-the-fly data-driven models to correct possible deviations between measurements and physics-based model predictions. This paper is focused on the computation of stable, fast and accurate corrections in the Hybrid Twin framework. Furthermore, regarding the delicate and important problem of stability, a new approach is proposed, introducing several sub-variants and guaranteeing a low computational cost as well as the achievement of a stable time-integration.

Alberto Badias, Iciar Alfaro, David Gonzalez et al.

We propose a new methodology to estimate the 3D displacement field of deformable objects from video sequences using standard monocular cameras. We solve in real time the complete (possibly visco-)hyperelasticity problem to properly describe the strain and stress fields that are consistent with the displacements captured by the images, constrained by real physics. We do not impose any ad-hoc prior or energy minimization in the external surface, since the real and complete mechanics problem is solved. This means that we can also estimate the internal state of the objects, even in occluded areas, just by observing the external surface and the knowledge of material properties and geometry. Solving this problem in real time using a realistic constitutive law, usually non-linear, is out of reach for current systems. To overcome this difficulty, we solve off-line a parametrized problem that considers each source of variability in the problem as a new parameter and, consequently, as a new dimension in the formulation. Model Order Reduction methods allow us to reduce the dimensionality of the problem, and therefore, its computational cost, while preserving the visualization of the solution in the high-dimensionality space. This allows an accurate estimation of the object deformations, improving also the robustness in the 3D points estimation.

Quercus Hernandez, Alberto Badias, David Gonzalez et al.

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.

Quercus Hernández, Alberto Badias, David Gonzalez et al.

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 so-called General Equation for the Non-Equilibrium Reversible-Irreversible Coupling, GENERIC [M. Grmela and H.C Oettinger (1997). Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys. Rev. E. 56 (6): 6620-6632]. 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 include conservative as well as dissipative systems, discrete as well as continuous ones.