Victor Champaney
email: victor.champaney@ensam.eufranciscochinestafrancisco.chinesta@ensam.eu

Francisco Chinesta

Elias Cueto
email: ecueto@unizar.es

Engineering empowered by physics-based and data-driven hybrid models: A methodological overview

Keywords: Smart manufacturing, Physics-based modelling, Model order reduction, PGD, Data-driven modelling, Artificial intelligence, Hybrid twins, Diagnosis and prognosis

Smart manufacturing implies creating virtual replicas of the processing operations, taking into account the material dimension and its multi-physics transformation when forming processes operate. Performing efficient, that is, online accurate predictions of the induced properties (including potential defects) of the formed part (to optimally control the process parameters) needs moving beyond usual offline simulation based on nominal models, and proceeds by assimilating data. This will serve, from one side, to keep the model calibrated, and from the other, to enrich the model and its associated predictions, to avoid bias, to improve accuracy or for performing online diagnosis, by advertising on preventive maintenance. For all these purposes, a new alliance between physics-based and data-driven modelling approaches seems a very valuable route for empowering engineering in general, and smart manufacturing in particular. The present paper revisits the main methodologies involved in the construction of the component or system Hybrid Twins.

Introduction

The main aim of engineering is moving from the product itself to the management of its performance all along its operational life. Even if the main aim of engineering was always the decision making, in virtue of accurate diagnosis and prognosis, it was not possible in the past due to the limitation in monitoring large components or systems. The irruption of sensors, communication, data-analytics, ... in all the domains of science, technology and society, within the so-called Internet of Things (IoT), originated major changes in engineering practices.

The last century engineering-very successful from multiple points of view-employed models expected to represent the physical system itself (which we will refer to as nominal models), calibrated by using data appropriately collected using sophisticated testing machines. Then, several loading scenarios, expected spanning the one that the component will experience during its real life, were applied to obtain as main output the component or system performances.

When the mathematical models became too complex for envisaging their analytical solution (within an almost analogical world) computers came to rescue for efficiently solving them. Despite of the numerous simplifying hypotheses involved in the model derivation, in its numerical treatment as well as in the definition of the loadings considered for the performances analyses, the success stories were abundant and some of them simply impressive, with most of the XX century technology as definitive proof. The considered engineering workflow is sketched in Fig. 1(left).

However, at the beginning of the third millennium, as just commented, managing the performance seems a real opportunity. Today when buying an electrical drill, we are in fact buying a good quality hole; nowadays engineering is more concerned by the number of hours of fly than by the aircraft engines themselves, ... This is the new face of the new performances-based engineering.

Engineering is concerned by the real system at the present time, with the knowledge of the loading that it experienced during its past life (fully described by the collected data). Thus, from all this data and knowledge we would like to make efficient diagnosis (at the present time) and prognosis (by projecting the present knowledge into the future) for anticipating and taking the right decision at the right time. The workflow of this new engineering is sketched in Fig. 1

(right).

The domain in which engineering operates is expanding, expecting to address systems the more and more large, complex, and involving the more are more variability and uncertainty. As just mentioned, engineering is not only addressing the design of components or systems, but monitoring them lo learn from their functioning, to make efficient (accurate and fast) diagnosis and prognosis, for the optimal decision making. To make all this possible, we need new techniques able to predict very accurately and very quickly complex system responses.

The two main words have been introduced, in a very simple and natural way: fast and accurate. In the past these two words were almost immiscible. When searching for fast responses models were degraded, simplified, ... and an amount of accuracy definitively lost. When searching for accuracy, models were kept rich enough, but then, the responses had to wait, or very powerful computers were used, but the last solution is not attainable in a context of democratizing simulation for rendering it accessible to small industries.

The two existing paradigms, the one of the past, mainly focused on physics and analogical knowledge, and the more recent and digital one, the one based on the employ of data manipulated by advanced artificial intelligence techniques, both present their inherent limitations. Both paradigms are sketched in Fig. 2.

As it can be noticed from Fig. 2, the physics-based modelling framework produces responses that approximate quite well the real ones, as soon as such an accurate model exist. The main difficulties when considering such a physics-based approach are: (i) the precision of the model itself; (ii) the impact that variability and uncertainty have in the predictions; and (iii) the computing time required for solving the complex and intricate mathematical models.

On the other hand, the data-driven framework is not fully satisfactory, because as it can be noticed in that figure, even if the data (assumed noise-free and symbolized by the blue small squares) represent well the reality, in between, the use of a low degree interpolation (in this image piecewise linear) produces a quite poor approximation of the real system response. Of course, by increasing the number of collected data, one could expect approximating the real solution better, however, data is not always Fig. 1 The past (left) versus the present (right) engineering practices Fig. 2 The physics-based (left) versus the data-driven (right) modelling paradigms simple to collect, not always possible to have access to it, and in all the cases collecting data is expensive (cost of sensors, cost of communication and analysis, ...). Equipping a very large industrial or civil infrastructure with millions of sensors to cover all its spatial dimension seems simply unreasonable. Moreover, even when the solution will be well approximated, two difficulties persist: (i) the solution explainability, compulsory to certify solutions and decisions; and (ii) the domain of validity, because even data can be interpolated in a quite safe manner, extrapolating them becomes extremely risky.

In view of the limitations of both existing frameworks, a gateway consists of allying both of them to conciliate these two targets: fast and accurate. The hybrid paradigm, sketched in Fig. 3, is a valuable and appealing option. The hybrid paradigm considers the reality expressible from the addition of two contributions: the existing knowledge (the state-of-the-art physics-based models or any other kind of knowledge-based models) and the part of the reality that our models ignore, the so-called ignorance (also called deviation, gap, discrepancy, ...). As can be noticed in Fig. 3, the model represents quite accurately, even if not perfectly, the reality.

In some cases, the discrepancy becomes simpler in the sense of cheaper to describe (less data) or to explain (compared with a fully data-driven approach), with most of its richness captured and explained by the physics-based model. Thus, as in transfer learning, or reinforced transfer learning, we try to approximate a richer behavior from another close enough and well stablished. This hybrid approach that combines the use of physics and data, can be applied in different ways, some of them reported below:

-In some cases, when the model captures most of the solution complexity, the correction must describe a discrepancy that could exhibits much smaller nonlinearities, as was the case treated in [START_REF] Sancarlos | From ROM of electrochemistry to AI-based battery digital and hybrid twin[END_REF][START_REF] Sancarlos | Learning stable reduced-order models for hybrid twins[END_REF], where the same amount of data performed better within the hybrid than within the fully data-driven framework. -Sometimes the physics-based model operates very accurately in a part of the domain, whereas the nonlinear behavior localizes in a very small region that can, in that case, be captured by a data-driven learned model, as considered in [START_REF] Reille | Learning data-driven reduced elastic and inelastic models of spot-welded patches[END_REF] for addressing the inelastic behavior of spot-welds. -When considering constitutive modeling of materials, different frameworks exist:

-The first is based on the collected data, complemented with first principles (and their associated variational formulations), as considered in the seminal work of Ortiz [START_REF] Kirchdoerfer | Data-driven computational mechanics[END_REF] with a lot of extensions. -A second approach considers the collected data to lie on a manifold embedded into the higher-dimensional behavior space [START_REF] Ibanez | Data-driven non-linear elasticity. Constitutive manifold construction and problem discretization[END_REF][START_REF] Ibanez | Numerical experiments on unsupervised manifold learning applied to mechanical modeling of materials and structures[END_REF][START_REF] Latorre | What-you-prescribe-is-whatyou-get orthotropic hyperelasticity[END_REF]. -A third approach includes a number of quite general rules representing the material description, for constructing the so-called constitutive manifold that intimately embraces data and existing knowledge [32]. -Other physics-informed approaches, within a thermodynamical dissipative setting [START_REF] Gonzalez | Thermodynamically consistent data-driven computational mechanics[END_REF]. -Within the augmented rationale (or hybrid paradigm) the real behavior can be assumed represented by a first order one (calibrated at best from the available data) complemented by an enrichment (or correction) filling the gap between the collected data and the predictions obtained from the chosen and calibrated model [START_REF] Gonzalez | Learning corrections for hyper-elastic models from data[END_REF].

-The hybrid modeling can also transfer the existing knowledge slightly outside its domain of applicability with small amount of collected data, as performed in [START_REF] Moya | Digital twins that learn and correct themselves[END_REF] for correcting the usual structural beam models. -Sometimes the discrepancy concerns an imperfect alignment of the solution between the prediction and the measures. That discrepancy seems very high when evaluating it at each location, however, a small transport allows aligning both solutions. Optimal transport is very suitable in these situations, where the hybrid model consists of the usual nominal model enriched from a parametric correction formulated in an optimal transport setting as described in [START_REF] Torregrosa | Surrogate Parametric Metamodel based on Optimal Transport[END_REF][START_REF] Torregrosa | Hybrid twins based on optimal transport[END_REF]. -In [START_REF] Ibanez | Hybrid Constitutive Modeling: Data-driven learning of corrections to plasticity models[END_REF] a correction of a given yield function was proposed and obtained from the deviation between the results predicted by using it and the measures obtained at the structure level. Being the correction much simpler it can be described from few approximation functions, much less that the ones needed for approximating the resulting yield function. -Finally, when addressing processing and performances, the hybrid formulation, of different granularities, can exhibit advantages (amount of data, ability to explain, knowledge transfer, ...), at the heart of the digital twin developments [START_REF] Argerich | Empowering Design Based on Hybrid Twin: Application to Acoustic Resonators[END_REF][START_REF] Casteran | Application of Machine Learning tools for the improvement of reactive extrusion simulation[END_REF]12,[START_REF] Ghanem | Probabilistic learning and updating of a digital twin for composite material systems[END_REF][START_REF] Kapteyn | From physics-based models to predictive digital twins via interpretable machine learning[END_REF]47].

Moreover, as bonus, the major contribution of the hybrid model, the one based on the state-of-the-art physics-based model, can be explained and is nowadays certifiable, fact that allows leaving the blackbox image characteristic of fully data-driven models, to obtain a light-gray box. It is important also to remark, that even if it seems conclusive the fact that data is helping the models to be more accurate, the existing knowledge also transform the big-data paradigm into a smarter one, where thanks to the existing knowledge, one can conceive the data to be collected and the optimal locations and time to perform the measurements. Despite of the undoubtable advantages in using the hybrid framework, three difficulties persist:

On the description of complex data and the associated metrics

Manipulating data for clustering or classification purposes is very common. It is expected that as soon a collected data is found belonging to a certain group, it will share some properties that characterize the members of the group, that is, proximity implies similar responses (in a certain sense).

The main issue is precisely how evaluating the proximity between data, because such proximity implies the use of adequate metrics, and sometime those metrics are not straightforward defined. This is for example the case when manipulating data associated with manufacturing processes described by their processing cards, consisting of the name of the employee involved in the operation, the designation of the employed materials, the temperature of the oven used for curing the part and the processing time. In that case, comparing two parts, from their respective processing cards, becomes a tricky issue. In these circumstances the comparison metrics itself must be learned from the existing training data, as is the case when using decision-trees (or its random forest counterpart) [START_REF] Breiman | Random forests[END_REF][START_REF] Kirkwood | Decision tree primer[END_REF], code-to-vector [START_REF] Argerich | Code2vect: An efficient heterogenous data classifier and nonlinear regression technique[END_REF] or neural networks [START_REF] Goodfellow | Deep learning[END_REF].

In other cases, the data exhibits a huge topological content, as it is the case when processing time-series or images of complex microstructures (foams, ...). The question is again how comparing two time-series or two images, that seem similar in a certain intuitive sense, but are impossible to match point to point. In many circumstances, techniques that operate by trying to align the data, like DTW [START_REF] Muller | Information retrieval for music and motion[END_REF][START_REF] Senin | Dynamic time warping algorithm review. Technical report 44. Tibshirani R[END_REF], fail to accomplish the task. Thus be most valuable route seems the one of extracting valuable statistical descriptors, to be then used in unsupervised clustering, supervised classification or for modeling purposes, as inputs of advance nonlinear regressions.

One of the most appealing techniques for performing that extraction consists of using the so-called topological data analysis (TDA). It has been successfully considered in our former works for addressing complex mesostructures [START_REF] Yun | Nonlinear regression operating on microstructures described from Topological Data Analysis for the real-time prediction of effective properties[END_REF], time-series [START_REF] Frahi | Empowering advanced driver-assistance systems from topological data analysis[END_REF], rough surfaces [START_REF] Frahi | Tape surfaces characterization with persistence images[END_REF] and shapes [START_REF] Frahi | Empowering advanced parametric modes clustering from topological data analysis[END_REF], with the aim of classifying and also constructing robust regressions expressing properties or performances from the input data expressed from its topological description.

To briefly summarize the main steps concerned by the application of TDA, we consider below its application in the analysis of time series and images.

Time series

We consider the time-series Y = {y 1 , y 2 , ..., y k , ...} , with y k ≡ y(t k ) (or y k ≡ y(x k ) when considering a profile y para- metrize by the coordinate x). Now, the local maximums and local minimums are identified, and taking into account its respective local-minimum to local-maximum proximity, matched, giving rise to the pairs (m p , M p ) , p = 1, ..., , where the lower case m is used for the local minimum and the capital letter M for the maximum. In the TDA terminology, minimum and maximum pairs are referred by birth and death respectively.

When reporting the pairs (m p , M p ) into a 2D diagram, with the birth in the abscissa axis and the death if the ordinate axis, the points will constitute the so-called persistence diagram.

The persistence diagram can be transformed into the socalled lifetime diagram, by representing in the ordinate axis the lifetime, i.e. M pm p , ∀p , instead of the death M p . Now, by associating to each point in the lifetime diagram a bivariate normal distribution and by embedding the lifetime diagram into a lager 2D domain covering (up-to a certain tolerance) the support of all the regularized points in the life-time diagram, after some technical manipulations, the so-called persistence image is obtained. It can viewed as a regularized form of the lifetime diagram.

The greatest merit of such a representation is its inherited invariance, that implies that different time series with similar topological content (similar persistence diagram) and even if they seems very different when expressed in the time or x-domain, their persistence image will be almost the same. Adequate metrics can be applied on both, the persistence diagram and the persistence images, mostly on the persistence images, making possible comparing, classifying and using them for modelling (regressions) purposes.

Complex miscrostructures

In the case of complex microstructures described from 2D images, a set of structured or unstructured points is distributed on the analyzed phase (e.g. the material when assuming a cellular material, like a foam). Now, the evolution of points, edges and triangles is evaluated with respect to an increasing distance, is evaluated.

When this distance starts with a null value only points exist. Then, when the distance increases points connect, to create edges. When the edges create a closed loop (a hole), a topological feature appears, and the current value of the distance represents the birth of the feature. Then, for certain value of the distance the hole disappears, fully filled by the just created triangles inside. This value correspond with the death. The birth and the death represents a point to be reported into the persistence diagram.

Then, moving from the persistence diagram to the lifetime diagram and finally to the persistence image, follows the same rationale previously described for the time series.

Physics in real-time

When looking for an approximation of the solution u(x, t) of a given partial differential equation -PDE-expected governing a particular physical phenomenon, here assumed scalar without loss of generality, the standard finite element method considers the approximation where U i denotes the value of the unknown field at node i, x i , and N i (x) represents the its associated approximation func- tion-shape function in the finite element method (FEM) terminology-. Here, refers to the number of nodes considered to approximate the field in the domain where the problem is defined.

This approximation results in an algebraic problem of size in the linear case, or the solution of many of them in the general transient and nonlinear cases. In order to alleviate the computational cost, model order reduction techniques have been proposed and are nowadays intensively used.

When considering POD-based model order reduction [11], a learning stage allows extracting the significant modes i (x) that best approximate the solution. Very often a reduced number of modes ( ≪ ) suffices to approxi- mate the solution of problems similar to the one that served to extract the modes at the learning stage. In other words, while finite element shape functions are general and can be employed in virtually any problem, the reduced-order basis is restricted to the domain where the learned process was accomplished.

By expressing the solution u(x, t) onto the reduced basis { 1 (x), … , (x)} the resulting discrete problem will now require the solution of a linear system of equations of size , instead of size , which is the actual size of the finite element solution. This often implies impressive savings in computing time. Addressing nonlinear models requires the use of specific strategies to ensure solution efficiency [START_REF] Chaturantabut | Nonlinear model order reduction via discrete empirical interpolation[END_REF].

(1) u(x, t) = ∑ i=1 U i (t)N i (x), (2) u(x, t) ≈ ∑ i=1 i (t) i (x),
Equation 1 or 2 involve a finite sum of products composed by time-dependent coefficients multiplied by space functions. These space function are the well-known finite element shape functions when no prior knowledge about the structure of the problem exists, or the modes extracted by applying POD.

A generalization of this procedure consists in assuming that space functions are also unknown. This makes it necessary to compute both time and space functions, on the fly. Thus, the resulting approximation reads Since the pairs of space and time functions in Eq. 3 are unknown, their determination will define a nonlinear problem. Obviously, it will require some form of linearization. This linearization procedure has been studied in some of the author's former works [START_REF] Chinesta | Parametric PGD based computational vademecum for efficient design, optimization and control[END_REF][START_REF] Chinesta | The proper generalized decomposition for advanced numerical simulations[END_REF]. The final approximation, Eq. 3, will require the solution of about problems, with ≪ and ∼ . Degenerate geometries (beams, plates, shells, layered domains such as composite materials) are specially well suited for a space domain separation. If the domain can be decomposed as = x × y × z , the solution u(x, y, z) could be approximated in turn by a separated representation of the type [START_REF] Chinesta | The proper generalized decomposition for advanced numerical simulations[END_REF] which is specially advantageous, since it gives rise to a sequence of one-dimensional problems instead of the typical three-dimensional complexity. For some geometries, like plates or shells, in-plane/out-of-plane this separated representation becomes specially interesting, where the obtained complexity of the problem is roughly the typical of a two-dimensional problem, i.e., the calculation of in-plane functions X i (x, y).

A very interesting case is that of space-time-parameter separated representations [START_REF] Chinesta | Parametric PGD based computational vademecum for efficient design, optimization and control[END_REF]. In this framework a so-called computational vademecum (also known as abacus, virtual charts, nomograms, ...) can be developed so as to provide a sort of computational response surface for the problem at hand, but without the need for a complex sampling in high dimensional domains. It has been successfully employed in problems like simulation, optimization, inverse analysis, uncertainty propagation and simulation-based control, to cite a few [START_REF] Chinesta | Parametric PGD based computational vademecum for efficient design, optimization and control[END_REF]. Once constructed off-line, this sort of response

(3) u(x, t) ≈ ∑ i=1 T i (t)X i (x). (4) u(x, y, z) ≈ ∑ i=1 X i (x)Y i (y)Z i (z), (5)
u(x, y, z) ≈ ∑ i=1 X i (x, y)Z i (z),
surface provides results under very stringent real-time constraints-in the order of milliseconds-by just invoking this response surface instead of simulating the whole problem. Thus, when the unknown field is a function of space, time and a number of parameters 1 , … , , the subsequent sepa- rated representation could be established as

The standard, intrusive, PGD constructor

We assume the generic model governing the evolution of the field under consideration u(x, t)

where L(•) represents a generic linear or nonlinear differ- ential operator, F(•) the so-called forcing term, and is a parameter (here a single parameter for the sake of simplicity) that could affect the domain in which the problem is defined, the physical model itself or the forcing term. Thus, it is expected that the solution will depend on the considered value of the parameter , i.e. u(x, t; ).

The Proper Generalized Decomposition (PGD) is based on the solution separated representation, originally proposed for defining non-incremental transient solutions [START_REF] Ladeveze | New Approaches and Non-Incremental Methods of Calculation 32[END_REF]. In the parametric setting, the PGD proceeds by assuming a fully separated representation of the problem solution, where parameters are assumed extra-coordinates [START_REF] Chinesta | Parametric PGD based computational vademecum for efficient design, optimization and control[END_REF]. Thus, the parametric solution approximation u (x, t, ) reads with x ∈ , t ∈ T and ∈ I.

To compute the different unknown functions involved in the separated representation Eq. 8 the usual weighted residual form is extended according to Within a Galerkin framework, when looking for the functional product m, the trial and test functions to be employed within the integral form (9), read respectively and ( 6)

u(x, t, 1 , … , ) ≈ ∑ i=1 X i (x)T i (t) ∏ j=1 M j i ( j ). (7) R(u(x, t); ) ≡ L(u(x, t); ) -F(x, t; ) = 0, in (8) u (x, t, ) = ∑ i=1 X i (x)T i (t)M i ( ), (9)
∫ ×T×I u * (x, t, )R(u(x, t, )) dx dt d = 0. (10) u m (x, t, ) = m-1 ∑ i=1 X i (x)T i (t)M i ( ) + X q (x)T q (t)M q ( ) =u m-1 (x, t, ) + X m (x)T m (t)M m ( ),
The separated representation constructor deeply described in [START_REF] Chinesta | The proper generalized decomposition for advanced numerical simulations[END_REF] proceeds by using an alternate direction fixed point algorithm that computes the unknown function at the enrichment iteration m: X m (x) from T m (t) and M m ( ) taken at the previous iteration, T m (t) from X m (x) and M m ( ) , and finally M m ( ) from X m (x) and T m (t) . The iteration continues until reaching the fixed point, and then the next functional product, X m+1 (x)T m+1 (t)M m+1 ( ) , is considered.

The main difficulties in applying the just described procedure are the necessity of performing an affine decomposition of the problem residual involved in Eq. 9 for making possible the sequential calculation of each one of the functions involving the corresponding problem coordinate. Such affine decomposition is not direct, mainly in the case of nonlinear models [START_REF] Aguado | Tensor representation of on-linear models using cross approximations[END_REF][START_REF] Chinesta | Parametric PGD based computational vademecum for efficient design, optimization and control[END_REF]. The other difficulty is related to the procedure intrusiveness that makes difficult its use in tandem with usual commercial software.

Non-intrusive PGD constructor

To overpass the just referred difficulties, one option consists in constructing metamodels (also known as surrogates, response surfaces, virtual charts or vademecums).

The construction of these metamodels can be performed by considering a quite simple workflow: (i) defining a sampling of the parametric space; (ii) computing a high fidelity solution for each parameters choice; (iii) using an adequate regression for extending the solution known at the points in the sampling everywhere in the parametric space.

There are many alternatives for performing the first and third tasks, trying to conciliate: (i) a sparse and very reduced sampling, considering the best sampling points location, and with the number of points roughly linearly scaling with the dimension of the parametric space; (ii) the high fidelity solutions post-compression by extracting first and using then (by projection) reduced bases; (iii) rich enough approximation bases while avoiding overfitting, based on the use of sparse regularizations for enforcing parsimony [START_REF] Brunton | Discovering governing equations from data by sparse identification of nonlinear dynamical systems[END_REF] (elastic-net, ridge, lasso, ...); (iv) using orthogonal basis for evaluating sensibilities in a direct manner; and (v) efficiently addressing the high-dimensional spaces induced by the multi-parametric models, where the use of separated representations are specially suitable ... [START_REF] Borzacchiello | Non-intrusive sparse subspace learning for parametrized problems[END_REF][START_REF] Ibanez | A multi-dimensional datadriven sparse identification technique: the sparse Proper Generalized Decomposition[END_REF][START_REF] Sancarlos | PGD-based advanced nonlinear multiparametric regressions for constructing metamodels at the scarce-data limit[END_REF]44].

In what follows the Sparse Subspace Learning (SSL) and the sparse PGD (sPGD) constructors, widely employed later, will be revisited.

(11)

u * (x, t, ) = X * (x)T m (t)M m ( ) + X m (x)T * (t)M m ( ) + X m (x)T m (t)M * ( ).

Sparse Subspace Learning

Again for the sake of simplicity, we assume that only one parameter is involved in the model, ∈ I ≡ [ min , max ] . The parametric solution u(x, t, ) is searched in the sepa- rated form SSL [START_REF] Borzacchiello | Non-intrusive sparse subspace learning for parametrized problems[END_REF] consists first in choosing a hierarchical basis of the parametric domain. The associated collocation points (the Gauss-Lobatto-Chebyshev) and the associated functions will be noted by:

( j i , j i ( ))
, where indexes i and j refer to the i-point at the j-level.

At the first level, j = 0 , there are only to points, 0 1 and 0 2 , that correspond to the minimum and maximum value of the parameters that define the parametric domain, i.e. 0 1 = min and 0 2 = max . If we assume that a direct solver is available, i.e., a computer software able to compute the transient solution as soon as the value of the parameter has been specified, these solutions read u 0 1 (x, t) = u(x, t, = 0 1 ) and u 0 2 (x, t) = u(x, t, = 0 2 ) respectively. Thus, the solution at level j = 0 could be approximated from that in fact consists of a standard linear approximation since at the first level, j = 0 , the two approximation functions read 0 1 ( ) = ( 0 2 -)∕( 0 2 -0 1 ) and 0 1 ( ) = ( -0 1 )∕( 0 2 -0 1 ) , respectively.

At level j = 1 there is only one point located just in the middle of the parametric domain, i.e. 1 1 = 0.5( min + max ) , being its associated interpolation function 1 1 ( ) . It defines a parabola that takes a unit value at = 1 1 and vanishes at the other collocation points of level j = 0 , 0 1 and 0 2 in this case. The associated solution reads u 1 1 (x, t) = u(x, t, = 1 1 ). This solution contains a part already explained by the just computed approximation at the previous level, j = 0 , expressed by Thus, we can define the so-called surplus as from which the approximation at level j = 1 reads

u(x, t, ) ≈ ∑ i=1 X i (x, t)M i ( ). u 0 (x, t, ) = u 0 1 (x, t) 0 1 ( ) + u 0 2 (x, t) 0 2 ( ), u 0 (x, t, 1 1 ) = u 0 1 (x, t) 0 1 ( 1 1 ) + u 0 2 (x, t) 0 2 ( 1 1 ). ũ1 1 (x, t) = u 1 1 (x, t) -u 0 (x, t, 𝜇 1 1 ), (12) u 1 (x, t, 𝜇) = u 0 (x, t, 𝜇) + ũ1 1 (x, t)𝜉 1 1 (𝜇).
The process continues by adding surpluses when going-up with the hierarchical approximation level. An important aspect is that the norm of the surplus can be used as a local error indicator, and then when adding a level does not contribute sufficiently, the sampling process can stop. The computed solution, as noticed in Eq. 12, ensures a separated representation. However, it could contain too many terms. In that circumstances a post-compression takes place by looking for a more compact separated representation [START_REF] Chinesta | The proper generalized decomposition for advanced numerical simulations[END_REF].

When the model involves more parameters 1 and 2 , here noted for the sake of notational simplicity and , the hierarchical 2D basis, defined in the parametric space ( , ) is composed by the cartesian product of the collocations points and the tensor product of the approximation bases.

Thus, the first level j = 0 , is composed by the four points:

with the associated interpolation functions When moving to the next level, j = 1 , the collocation points and approximation functions result from the combination of the zero-level of one parameter and the first level of the second one, i.e., the points are now: ( 0 1 , 1 1), ( 0 2 , 1 1) and

( 1 1 , 0 1 ), ( 1 1 , 0 2 ) .
In what concerns the interpolation func- tions they result from the product of the zero level in one coordinate and the level one in the other. It is worth noting that the point ( 11 , 1 1 ) and its associated interpolation func- tion is in fact a term of level j = 2.

The main issue when using the SSL is that even when limiting the approximation to the first level (the zero level), if dimensions are involved, the number of collocation points becomes 2 , that limits the use of the SSL when both, the number of parameters, and/or the degree to be considered (the so-called level) increase. For alleviating this issue the sPGD and its regularized variants were proposed, and will be summarized below.

From the sparse PGD to its regularized variants: rsPGD and s2PGD

For the the sake of simplicity we consider two parameters, again noted by and , and a field that depend on both them, u(x, t, , ) , with n t known solutions associated with n t dif- ferent choices of the parameters, i.e. u(x, t, i , i ) ≡ u i (x, t) , i = 1, ..., n t .

In general these solutions are expressed in a discrete manner, in the form of a matrix, where each column contains the solution at each node (rows) at a given time step. Thus, the discrete form of the solution u(x, t, i , i ) ≡ u i (x, t) is

( 0 1 , 0 1 ), ( 0 2 , 0 1 ), ( 0 2 , 0 2 ), ( 0 1 , 0 2 ), 0 1 ( ) 0 1 ( ), 0 2 ( ) 0 1 ( ), 0 2 ( ) 0 2 ( ), 0 1 ( ) 0 2 ( ).
noted i , with the component i rs referring to the solution at node x r at time t s , for the parameters choice ( i , i ) , i.e. i rs ≡ u(x r , t s , i , i ).

However, finding a parametric regression of each component of the discrete solution, leading to ( , ) , is too expensive. For this reason, more than working with the discrete solution itself, reduced space and time bases are extracted: k , k = 1, ..., , and l , l = 1, ..., . Then, each discrete solution i can be expressed as where = ( 1 ... ) , = (T 1 ... T L ) and , whose size is × , is obtained by minimizing ‖ ii ‖. Now, knowing i pq , i.e. each component of it for each choice i, i = 1, ..., n t , of the parameters, ( i , i ) , i pq ≡ pq ( i , i ) , the more general parametric expression pq ( , ) is searched.

For the sake of notational simplicity, we will describe in what follows the construction of regressions of a generic scalar f that depends on the couple of parameters ( , ) , with f i ≡ f ( i , i ) known at the n t points in the sam- pling. Here, the scalar f represents each component pq , p = 1, ..., and q = 1, ..., .

The goal is therefore to find a function f M ( , ) , expressible as a finite sum of terms, according to able to approximate the known data

f i ≡ f ( i , i ) , i = 1, ..., n t .
Even if in the 2D case addressed here (with only two parameters) the separation of variables is not compulsory, we would like to propose a technique general enough and able to operate in highly multi-parametric settings.

At iteration m the algorithm looks for the update

G m ( )H m ( ) such that f m ( , ) = f m-1 ( , ) + G m ( )H( ) .
For computing the searched functions, they are first approximated as G m ( ) = ,T m ( ) m and H m ( ) = ,T m ( ) m , where m represents the basis considered for approximating the m-mode depending on the -parameter, being m the associated weights, and similarly for the other direction ( -parameter): m and m .

The update comes from the minimization problem:

that by using the notation below:

(13) i ≈ i , (14) f
M ( , ) = ∑ j=1 G j ( )H j ( ), (15)
G m ( )H m ( ) = argmin (G m ( )H m ( )) * n t � i=1 ‖f i -f m-1 + (G m ( )H m ( )) * ‖ 2 2 ,
results in the two problems:

that are solved iteratively until reaching the fixed point. This schema constitutes the heart of the sPGD constructor. However, when combining rich approximation bases m ( ) and m ( ) , with data-sets n t not sufficiently rich (as it will be always the case when operating in highly multiparametric settings) the just described procedure produces overfitting.

To alleviating overfitting, an adaptive procedure was proposed in [START_REF] Ibanez | A multi-dimensional datadriven sparse identification technique: the sparse Proper Generalized Decomposition[END_REF] that consists in adapting the approximation bases, whose degree increases when advancing in the modal enrichment, that is, with the degree increasing with m.

Looking for a more versatile and automatic procedure, the ridge regularization was employed, giving rise to When looking for sparsity in order to employ extremely rich approximations while exploiting parsimony, like proposed in the Sindy [START_REF] Brunton | Discovering governing equations from data by sparse identification of nonlinear dynamical systems[END_REF], it is well known that the lower is the employed norm the more intense results the sparsity enforcement. Many times the L1-norm (the so-called Lasso regularization) becomes a good compromise between sparsity enforcement and computational efficiency, and when combining ridge and Lasso the so-called elastic net regularization results, that when combined with the separated representation constructor reads [START_REF] Sancarlos | PGD-based advanced nonlinear multiparametric regressions for constructing metamodels at the scarce-data limit[END_REF]

r = ⎛ ⎜ ⎜ ⎝ f 1 -f m-1 ( 1 , 1 ) ⋮ f n t -f m-1 ( n t , n t ) ⎞ ⎟ ⎟ ⎠ , = ⎛ ⎜ ⎜ ⎝ ,T m ( 1 )b m ,T m ( 1 )) ⋮ ,T m ( n t )b m ,T m ( n t )) ⎞ ⎟ ⎟ ⎠ , = ⎛ ⎜ ⎜ ⎝ ,T m ( 1 )a m ,T m ( 1 )) ⋮ ,T m ( n t )a m ,T m ( n t )) ⎞ ⎟ ⎟ ⎠ , ( 16
) a m = argmin a * m � ‖r - a * m ‖ 2 2 � , ( 17
) b m = argmin b * m � ‖r -b * m ‖ 2 2 � , (18)
a m = argmin a * m � ‖r - a * m ‖ 2 2 + ‖a * m ‖ 2 2 � , ( 19
) b m = argmin b * m � ‖r -b * m ‖ 2 2 + ‖b * m ‖ 2 2 � . ( 20
)
a m = argmin a * m � ‖r - a * m ‖ 2 2 + � (1 -)‖a * m ‖ 2 2 + ‖a * m ‖ 1 � � ,
Figure 4 summarizes the regression techniques usually employed to conciliate: (i) a sparse and very reduced sampling; (ii) rich enough approximation bases while avoiding overfitting, based on the use of sparse regularizations; (iii) using orthogonal basis for evaluating sensibilities in a direct manner; efficiently addressing the high-dimensional spaces induced by the multi-parametric models, where the use of separated representations are specially suitable.

Physics-aware artificial intelligence

Machine Learning (ML), a major protagonist of AI, is able to create predictive models from available or collected data, with an additional added value, the fact of providing that prediction in almost real-time. However, creating models from scratch, simply because the existing models based on physics were not accurate enough, is not the best choice. Creating a model based on data from scratch needs a lot of data, and in engineering and technology, data is synonym of cost, and sometimes the data collection also implies to consider an ethical dimension, to fulfill existing regulations, or must address technical difficulties.

In our work we advocate by the alliance between both (i) the former analogical world of knowledge and physicsbased models and (ii) the more recent digital world of data, manipulated by the more and more powerful (accurate, frugal and explainable -certifiable-) techniques of Artificial Intelligence.

This new alliance is materialized in the so-called Digital or Hybrid Twins, in which physics-based models are enriched, to decrease their intrinsic ignorance, in a pragmatic way, from the data representing the deviation between predictions and measurements. The hybrid paradigm not only allows reducing the amount of needed data that now is only expected describing the gap between the reality and the physics-based predictions, but also the ability of explaining the part of the model based on the existing physics or knowledge, and then facilitating the design or decision certification.

In order to reduce the amount of data to model the observable phenomena (when models do not exist or are too inaccurate) or for enriching the existing physics-based models within the hybrid paradigm previous introduced, physics-aware (also known as physics-informed) Artificial Intelligence (Machine Learning or Neural Networks) seems being the most appealing route.

* m � ‖r -b * m ‖ 2 2 + � (1 -)‖b * m ‖ 2 2 + ‖b * m ‖ 1 � � .
The so-called Physics Informed Neural Networks (PINN) [START_REF] Raissi | Physics informed deep learning (part I): data-driven solutions of nonlinear partial differential equations[END_REF][START_REF] Raissi | Physics informed deep learning (part II): data-driven discovery of nonlinear partial differential equations[END_REF] considers the approximation of the unknown function u(x, t) as a regression problem defined on an adapted neural network. Then, as soon as the physics is assumed fully known and adequately described by a partial differential equation, the derivatives involved in the differential operator can be applied on the NN and the residual nullity is enforced from the NN loss function.

Sometimes, everything concerning the physics is not fully known. If we consider an hyper-elastic material, the best option consists in learning the free energy by constructing a regression linking it to the state variables, in such a way that its derivatives leads to the constitutive equation. Then, the free energy is learned to be consistent with the collected data on the structural component, under the equilibrium constraints.

Similar procedures apply in the so-called Structure Preserving NN (also known as Thermodynamic Informed NN) where the free energy and the dissipation potential are computed in such a way that energy balance and entropy production are ensured [START_REF] Gonzalez | Thermodynamically consistent data-driven computational mechanics[END_REF][START_REF] Gonzalez | Learning corrections for hyper-elastic models from data[END_REF][START_REF] Hernandez | Deep learning of thermodynamics-aware reduced-order models from data[END_REF][START_REF] Hernadez | Learning non-Markovian physics from data[END_REF][START_REF] Moya | Learning slosh dynamics by means of data[END_REF][START_REF] Moya | Digital twins that learn and correct themselves[END_REF].

In those works, inspired from the GENERIC framework [START_REF] Grmela | Dynamics and thermodynamics of complex fluids. I. development of a general formalism[END_REF], the state evolution, ̇ , reads

(22) ̇ = ∇ Z H + ∇ Z S,
where the first term of the right-hand member represents the reversible evolution (Hamiltonian contribution) whereas the second one represents the dissipative contribution, with H and S the energy and the entropy respectively.

Learning matrices and , skew-symmetric the former, and symmetric and positive semi-definite the last, as well as both potentials H and S (subjected to some constraints: the Jacobi identities as well as the consistency conditions ∇ Z S = and ∇ Z H = ) is performed from the existing data concerning the state time evolution (t i ) , i = 1, 2, ... The learned model has very interesting properties, as are the ones related to energy conservation and positive dissipation, enabling stable and accurate time integrators.

The main issue found when learning such thermodynamic-aware models is not the regression implementation, but the choice of the variables in the state vector, issue discussed in [START_REF] Ibanez | Numerical experiments on unsupervised manifold learning applied to mechanical modeling of materials and structures[END_REF]. In very small systems the state variables are easily identified, however in large (continuous) systems such a choice is far of being trivial. In that case, many options exist. One among them consists of performing a dimensionality reduction. The use of most of manifold learning dimensionality reduction has as main drawback the unavailability of performing the inverse mapping for coming back from the reduced space to the departure one.

In the general nonlinear case, an appealing alternative consists in the use of NN-based autoencoders [START_REF] Hinton | Autoencoders, minimum description length and Helmholtz free energy[END_REF], where encoding and decoding is learned, while models operate Fig. 4 Advanced nonlinear regressors combining different sparse regularization. They enable addressing nonlinear behaviors with a low amount of data, while avoiding overfitting, in the multi-parametric setting efficiently addressed by the separated representations at the heart of the so-called PGD [START_REF] Sancarlos | PGD-based advanced nonlinear multiparametric regressions for constructing metamodels at the scarce-data limit[END_REF] (and are learned) in a transparent way in the internal layer of reduced dimension where active reduced coordinates (the so-called latent variables or latent space) act. Autoencoders allow learning the best model representation, that in many cases, other than reducing the dimension of the model, allow reducing the model complexity (and then its nonlinearity) enabling the use of simpler and cheaper regression techniques operating at the level of the latent space, where sometimes linear regressions suffice.

Use case

The present section illustrates the just introduced concepts, methodologies and procedures, on a forming process usecase involving stamping, and more particularly multi-stage stamping, where the deviations, due to inevitable uncertainties, accumulate throughout the whole process compromising the geometrical tolerancing of the final formed parts.

In what follows we consider a two-stage stamping process, where the first stage, depicted in Fig. 5, imprints a spherical shape on the original planar metallic sheet, whereas the second one acts on the shape that resulted from the first-stage, to imprint a final cylindrical shape, as illustrated in Fig. 6.

Both stages involve three main parameters: (i) the clamping force F; (ii) the tool-sheet friction coefficient and (iii) the punch-sheet friction coefficient .

A parametric solution of the first stage was computed by using the non-intrusive PGD constructor previously introduced, operating on the solutions provided by the software PAM-STAMP (ESI Group) associated with the design of experiments associated with the SSL regression up-to level two.

The second stage operates on the shape that results from the first one. Thus, the parametric model of the second stage should include as parameters the ones describing the initial shape and its themomechanical state, representing the final state of the first stage.

As the final sate of the first stage is parametrized by the triplet (F I , I , I ) (where the superscript • I refers the first stage), one could expect having that triplet into the parametric space of the second-stage, and then expressing the nodal displacement according to II (F I , I , I , F II , II , II ). Such a parametric solution is depicted in Fig. 7.

Such a procedure faces three major difficulties: (i) the increase of the number of parameters when several stages are involved; (iii) the dependence of each stage model on the whole process; and (iii) the errors, inaccuracies and uncertainties accumulation from one stage to the next.

A way of increasing generality and accuracy consists in creating a generic parametric solution applicable to any stage. For that purpose, from the simulation of the whole process for different parameters, the final state of each stage is computed and the principal modes extracted by using the POD. Thus, in our case, from the available snapshots I i (F I i , I i , I i ) , i = 1, ..., , the most relevant POD modes j , j = 1, ..., , are extracted. In our case three modes ( = 3 ) suffice for describing the final state of the first stage. Accordingly, each snapshot can be expressed as where taking into account the orthonormality of the reduced basis j , the coefficients read More generally, any state resulting from the first stage could be expressed from ( 23)

I i ≡ I (F I i , I i , I i ) ≈ ∑ j=1 i j j , i = 1, ..., , (24)

Conclusion

This paper revisited the main methodologies employed in the hybrid engineering and hybrid twins, main protagonists of the incipient performance based engineering, where the necessity of predicting very fast and very accurately, while reducing the amount of needed data, leaded to the hybridization of physics-based models, efficiently manipulated thanks to the use of advanced non-intrusive model order reduction techniques, enriched, in view of the collected data, by using low-cost physics-aware artificial intelligence techniques.

Fig. 3

 3 Fig.[START_REF] Argerich | Empowering Design Based on Hybrid Twin: Application to Acoustic Resonators[END_REF] The Hybrid paradigm: the real systems is expressed from a physics-based model enriched from the data-driven model of the discrepancy

( 25 )

 25 I (F I , I , I ) ≈ ∑ j=1 j j .

Fig. 5 Fig. 6

 56 Fig. 5 Two-stage stamping. First-stage: (left) initial configuration and (right) almost final configuration

  (27) ̂ I ≈ α1 𝜙 1 + ... + α 𝜙 ,

Fig. 7 Fig. 8

 78 Fig. 7 Parametric solution II (F I , I , I , F II , II , II )

Acknowledgements Authors acknowledge the support of ESI Group from their chairs at the University of Zaragoza and at Arts et Metiers Institute of Technology.

This means that as considered in the POD by interpolation, one can express parametrically the coefficients j , i.e. j (F I , I , I ). Such a rationale becomes more general, and enables the solution of the second stage being expressed parametrically with respect to j , j = 1, ..., , and the process triplet (F II , II , II ) , i.e.

Other than gaining generality, because of the fact that Eq. 26 can be applied to any stage independently of the place that it occupies in the whole process, as soon as the reduced basis { 1 , … , } is able to express the incoming part, such a rationale enables also improving accuracy.

Within the fully physics-based rationale, the part resulting from the first stage, when enforcing the process parameters (F I , I , I ) , is expressed from with j = j (F I , I , I ) , j = 1, … , . If these coefficients are then inserted into the second stage parametric solution, given by Eq. 26, the computed solution will correspond approximately with II (F I , I , I , F II , II , II ). However, if the real part after the first stage, represented by ̂ I , differs from I , i.e. ‖̂ I -I ‖ > 𝜖 , a fully physics- based approach will propagate such a deviation to the stages that follow.

The hybrid paradigm aims at correcting the intermediate solutions. If we assume that the reduced basis continues spanning the real solution we can express it from ( 26) II ( 1 , ..., , F II , II , II ).

where αj = ̂ I ⋅ 𝜙 j . The real state ̂ I is rarely fully known, however, a partial knowledge suffices for determining the parameters αj . Few amount of data, scaling with , is in general sufficient.

Such a data-driven correction allows updating the prediction related to the second stage, ̂ II ( α1 , ..., α , F II , 𝜉 II , 𝜇 II ) instead of II ( 1 , ..., , F II , II , II ). In the numerical example addressed in the present section, as soon as the solution of the second stage is updated, the solution at the end for (F II , II , II ) results the one depicted in Fig. 8(left) where the damage exceed the maximum acceptable value. Thus, the control proceeds by changing the clamping force FII , action that leads to the solution depicted in Fig. 8(right) that exhibits an acceptable risk criterion.

Declarations

Conflicts of interest The authors declare that they have no conflict of interest.


MD a ; Bruno Pavy
email: pavy.bruno@wanadoo.fr

MD b ; Sophie Kubas

MD c ; Cécile Rocca

MD d ; Erick Merle

MD e ; Hélène Kerros

MD f ; Anne Tisseau

MD g ; Marie-Christine Iliou

PhD Anne Le Cunuder

Alain Cohen-Solal

MD, PhD François Carré

MD, PhD Alain Cohen- Solal

Evaluation of Maintained PhysicAl CaPacity 1-yr After Coronary Patient Cardiac Rehabilitation (EMAP): A FRENCH MULTICENTER STUDY

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Secondary prevention after a coronary event is essential, as lifestyle changes are difficult to implement [START_REF] Teo | Prevalence of a healthy lifestyle among individuals with cardiovascular disease in high-, middle-and low-income countries the prospective urban rural epidemiology (PURE) study[END_REF] . The benefits of cardiac rehabilitation (CR) to reduce cardiovascular morbidity and mortality are widely recognized in coronary artery disease (CAD) patients [START_REF] Anderson | Exercise-based cardiac rehabilitation for coronary heart disease: cochrane systematic review and meta-analysis[END_REF][START_REF] Pouche | Cardiac rehabilitation and 5-year mortality after acute coronary syndromes: the 2005 French FST-MI study[END_REF][START_REF] Doimo | Impact of ambulatory cardiac rehabilitation on cardiovascular outcomes: a long-term follow-up study[END_REF] . Multidisciplinary management with therapeutic patient education is particularly effective and recommended. This CR, modifies medical cardiovascular risk factors (hypertension, diabetes, hypercholesterolemia) and lifestyle (diet, smoking, sedentary/physical inactivity levels) [START_REF] Balady | Core components of cardiac rehabilitation/secondary prevention programs: 2007 update: a scientific statement from the American Heart Association Exercise, Cardiac Rehabilitation, and Prevention Committee, the Council on Clinical Cardiology; the Councils on Cardiovascular Nursing, Epidemiology and Prevention, and Nutrition, Physical Activity, and Metabolism; and the American Association of Cardiovascular and Pulmonary Rehabilitation[END_REF][START_REF] Thomas | Home-Based Cardiac Rehabilitation: A scientific statement from the American Association of Cardiovascular and Pulmonary Rehabilitation, the American Heart Association, and the American College of Cardiology[END_REF][START_REF] Ambrosetti | Secondary prevention through comprehensive cardiovascular rehabilitation: From knowledge to implementation. 2020 update. A position paper from the Secondary Prevention and Rehabilitation Section of the European Association of Preventive Cardiology[END_REF][START_REF] Kabboul | Comparative Effectiveness of the Core Components of Cardiac Rehabilitation on Mortality and Morbidity: A Systematic Review and Network Meta-Analysis[END_REF] .

Exercise training improves cardiorespiratory fitness (CRF), which is a major mortality prognostic marker. 1 metabolic equivalent of the task (MET) increase was associated with 8 to 35% reduction in risk of mortality (depending on the studied population characteristics) [START_REF] Harber | Impact of cardiorespiratory fitness on all-cause and disease-specific mortality: advances since 2009[END_REF] . In CAD patients, exercise training (anti-ischemic, antiarrhythmic, antithrombotic and psychologic effects) improves endothelial function and participates to CR benefits [START_REF] Franklin | Exercise-based cardiac rehabilitation and improvements in cardiorespiratory fitness: implications regarding patient benefit[END_REF] .

A large German multicenter study, whose CR modalities are similar to those used in France, observed a decrease in mortality after CR. This study reported a higher physical activity (PA) in the follow-up, but without any CRF evaluation [START_REF] Rauch | Short-term comprehensive cardiac rehabilitation after AMI is associated with reduced 1-year mortality: results from the OMEGA study[END_REF] . In a large UK community-based CR program with a 10 year mean follow-up, the baseline CRF which increased by mean 1.1 MET was the main survival rate predictor. However, the RAMIT study results with an average gain of only 0.5 MET after the CR program reported no effects on mortality [START_REF] Barons | Fitness predicts longterm survival after a cardiovascular event: a prospective cohort study[END_REF][START_REF] West | Rehabilitation after myocardial infarction trial (RAMIT): multi-center randomised controlled trial of comprehensive cardiac rehabilitation in patients following acute myocardial infarction[END_REF] .

The limits from self-reported data PA practice during phase III rehabilitation by patients are proven [START_REF] Kaminsky | Assessing physical activity as a core component in cardiac rehabilitation. A position statement of the American Association of Cardiovascular and Pulmonary Rehabilitation[END_REF] . Thus, it seems more appropriate to evaluate the patient CRF changes after the CR program to study the sustainability of its benefits.

The aim of this prospective study was to follow the evolution of the CRF level in CAD patients before, at the end, and one year after their CR program.

METHODS

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This multicenter study was carried out in 16 French CR centers which included at least 10 consecutive coronary patients. Multicenter study ethical approval was obtained with the personal protection committee (n°2015-S4). All participants provided written informed consent.

CRITERIA OF INCLUSION

All patients must have undergone within the previous 3 months an acute coronary syndrome (ACS), and / or percutaneous coronary intervention (PCI) revascularization, or coronary artery bypass graft (CABG). They had to participate in a CR stay including a physical training program with at least 5 sessions (inpatient or outpatient), performed two cycle ergometer exercise tests, and obtained a progression of at least 10 watts.

CRITERIA OF EXCLUSION

Patient refusal, misunderstanding, co-morbidity preventing PA, CR exclusion criteria according to the French cardiac rehabilitation guidelines [START_REF] Pavy | French Society of Cardiology guidelines for cardiac rehabilitation in adults[END_REF] .

PROTOCOL Cardiorespiratory fitness assessment:

Each patient had to perform 3 maximal or symptom-limited progressive exercise tests (ET) on a cycle ergometer. The exercise protocol usually started at 30 watts, with 10 watt increments every minute (or more to achieve an optimal 8-12 minutes test duration), until exhaustion or symptoms. The maximum validated workload was that of the last step maintained for at least 30 seconds. The initial ET (ET1) was performed prior to the physical training program, ET2 at the completion of the training program, and ET3 one year thereafter. The FRIEND formula was ACCEPTED MANUSCRIPT / CLEAN COPY chosen to estimate the CFR because it is more accurate than the ACSM formula [START_REF] Kokkinos | A new generalized cycle ergometry equation for predicting maximal oxygen uptake: The Fitness Register and the Importance of Exercise National Database (FRIEND)[END_REF] . To allow for accurate comparisons, the three ETs were performed using the same protocol while the patients were under medication.

Cardiac rehabilitation program:

The cardiac rehabilitation program included a therapeutic education program and at least 5 exercise sessions including continuous or interval-training endurance, resistance, and respiratory training. Physical training was practiced in groups, either residents (5 sessions per week) or outpatients (3 to 5 sessions per week), over a 3 to 7week period.

Data collection:

Demographic data gathered at the end of the CR and one year later concerned blood pressure, glycemia, LDL and HDL cholesterol levels. Diabetes was determined through positive blood tests (fasting blood sugar >1.26g/L) or current antidiabetic drug medication. Hypertension was confirmed if blood pressure >140/90 mmHg, or patient under current antihypertensive medication.

STATISTICAL ANALYSIS

Quantitative variables were expressed as mean ± standard deviation and qualitative variables as percentages. The comparison performed concerned the evolution of the parameter values observed during the 3 stages (pre-CR, post-CR and 1-year post-CR) of the study. Comparisons of the variables were conducted using analysis of variance (ANOVA) and Kruskal-Wallis according to distribution. We used Bonferroni's adjustment to identify potential differences among the individual three exercise tests. Then, we used independent t-test or Mann-Whitney U test for analysis of subgroups, as appropriate. The Chi-square test was used for the ACCEPTED MANUSCRIPT / CLEAN COPY percentages comparison and the Spearman test was used to assess the correlation between variables. In all cases the significance level was set at P < .05. Statistical analysis was performed using the software Stat-EL (www.adscience.fr)

RESULTS

POPULATION (see Figure 1)

From December 2015 to December 2016, in 392 eligible patients, 117 were excluded. 18 patients declined, 22 patients did not complete the CR (10 dropped out, 1 severe heart failure, 1 depressive syndrome, 10 without details), 16 did not perform two cycle ergometer ETs (7 with treadmill, 9 too tired), 7 patients did not progress (< 10 watts), 14 patients did not attend at least 5 training sessions, 15 patients were unable to participate (cognitive impairment, language barriers), 19 patients lived too far away, and 6 others were also excluded (two were already included in another protocol, 1 pericarditis, 1 severe ventricular dysfunction, 1 with a lifevest and 1 severe peripheral artery disease). Therefore, 275 patients were included in the study. Among them, 16 (6%) were lost during the 1-year follow-up: 2 non-cardiac deaths, 10 refused, and 4 did not attend the follow-up. Thus, 259 patients completed the whole study.

TRAINING PROGRAM

Total training session number was 17 ± 8, with an average duration of 122 ± 45 min per session.

The global training volume was 34.2 ± 16 h for an average duration of 5 ± 3.3 weeks (133 outpatients over a 6.9 week mean duration CR program and 126 inpatients for a 3.1 week mean duration program). The training volume was 41.7 ± 15 hours versus 26.2 ± 15 (P < .001) respectively for outpatients and inpatients. Medications were unchanged after one year.

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BASELINE DATA

Characteristics are summarized in Table 1.

The population was predominantly male (89%) with an average age of 60. 200 patients (77%) had ACS, including 149 ST-elevation myocardial infarctions. Their left ventricular ejection fraction was > 50%, 35-50%, and ≤ 35% in respectively 71, 24 and 5% of cases. Revascularization was performed on patients with one and two vessels CAD using PCI in 84% of cases, and on patients with three vessels CAD using CABG in 59% of cases.

CARDIOVASCULAR RISK FACTORS (Table 2)

In comparison with the initial evaluation, the body mass index (BMI) and the waist size were unchanged one year after the CR stay. The weight tended to increase (78.5 ± 13 versus 79.2 ± 13 kg, P = .05). Lipid balances showed a downward trend in LDL cholesterol (84.6 ± 39 vs 79.4 ± 28 mg / dL, P = .06) and an increase (12%) in HDL cholesterol (44.4 ± 13 vs 49.9 ± 14 mg / dL, P < .001). Regarding the 38 smoking patients, 18 had stopped during the follow-up, but 9 former smokers have resumed smoking.

EXERCISE TESTING DATA

Table 3 and Figure 2 summarize the ET data.

In comparison with ET1, the maximum workload increased by 29% ± 18 (P < .001) after CR and further increased by 5% ± 15 (P < .01) one year after CR.

Considering their maximal weight-indexed workload (watts/kg) and CRF (estimated METs), one year after the end of CR, 163 patients (63%) improved or maintained it (ET3 ≥ ET2), 73 (28%) decreased it (ET1 < ET3 < ET2) and 23 (9%) lost the objective (ET3 ≤ ET1) benefit of CR.

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The average CRF gain observed after the CR program was 1.1 ± 0.7 METs (22.5 ± 15 %) and 1-year after 1.3 ± 1.1 METs (26.4 ± 23 %) (Table 3).

Table 4 shows that while the absolute gain in watts or METs was similar regardless of the initial CRF, percent improvement increased with decreasing CRF.

Resting heart rate (HR) decreased between pre-CR and post-CR with no additional modification one year later. Peak HR and heart rate reserve (HRR; peak HR-resting HR) were improved between ET1 and ET2 (P < .01) as well as between ET2 and ET3 (P < .01).

The training volume during CR stay was moderately correlated with the post-rehabilitation workload improvement (r = 0.24, P < .001), but no correlation was found after one year (P = .92).

SPECIFIC POPULATIONS

The diabetic patients (n = 47) presented a lower CRF (4.7 ± 1.1 versus 5.4 ± 1.4 METs, P < .001) and a lower workload improvement post-CR (26 ± 31 vs. 30 ± 20%, P <.03) than the nondiabetic population. One year after CR, no difference was noted (3.2 ± 13 vs. 4.8 ± 15%, P = .62). Their HDL-cholesterol blood value did not significantly increase in the follow-up post CR (44.5 ± 14 vs. 48.4 ± 16 mg / dL, P = .08).

One year later, non-weaned smokers (n = 20), although younger (52 ± 7 versus 62 ± 10 years, P < .001), decreased their workload (-0.06 ± 0.13% versus 0.06 ± 0.16% for non-smokers and 0.06 ± 0.14% for weaned smokers, P < .004). Their HDL-cholesterol blood value did not increase one year post CR (47.8 ± 11 versus 45.6 ± 13 mg / dL, P = .41).

No correlation was found between age, gender, BMI, presence of hypertension or dyslipidemia, and change in CRF one year post CR.

OUTPATIENTS AND INPATIENTS

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The outpatients' CRF (n = 133) was higher than that of inpatients (n = 126), respectively 5.4 ± 1.4 vs. 5.1 ± 1.3 METs (P < .002) before the CR program, and 6.7 ± 1.6 vs. 6.1 ± 1.5 METs (P < .001) at the end of the CR program. Improvement was also more significant, 25 ± 17 vs. 20 ± 12% (P < .004) for inpatients. After one year, the CRF was similar (6.7 ± 1.7 vs. 6.5 ± 1.7

METs (P = .18) in both groups.

FOLLOW-UP

During the follow-up, 38 patients (15%) were hospitalized, including 13 (5%) in cardiology.

Five patients (1.9%) were revascularized including 2 ACS (0.7%), 1 atrial flutter ablation, 3 cardiac defibrillator and 2 pace-maker implantations.

DISCUSSION

This study analyses the early and medium-term (1 year) benefits of a CR program carried out in real life in French CR centers (in and outpatients) on the maximum workload and on the CRF of CAD patients. The global CRF gain observed and maintained after one year was 1.3 METs, which is consistent.

The design of our study, which made sure to use the same protocol during the 3 exercise tests, and the fact that they were all carried out with unchanged individual medication, allowed a reliable comparison of the results observed during these three tests.

The CRF improvement was slightly lower than the average value of 1.5 METs reported in a recent meta-analysis [START_REF] Sandercock | Changes in cardiorespiratory fitness in cardiac rehabilitation patients: A meta-analysis[END_REF] . The small discrepancy could be explained by the fact that the metaanalysis concerned outpatients and that their CRFs were evaluated on a treadmill. Moreover, the number of sessions was most often 36 during a 12 week duration. In our study, we also noted that outpatients benefited from a larger volume of training than inpatients, and presented ACCEPTED MANUSCRIPT / CLEAN COPY higher CRF improvement during the CR program. However, this benefit disappeared one year after the CR program. The CRF improvement we observed one-year post CR is in accordance with previous results observed with specific studies [START_REF] Boesch | Maintenance of exercise capacity and PA patterns 2 years after cardiac rehabilitation[END_REF][START_REF] Arrigo | Comparative outcome one year after formal cardiac rehabilitation : the effects of a randomized intervention to improve exercise adherence[END_REF] . Moreover, our results are consistent with those of the SAINTEX multicenter study comparing continuous training versus intervaltraining, which found a similar CRF stability one year after the CR program, as well as with the EU-CaRE register in older patients [START_REF] Pattyn | The long-term effects of a randomized trial comparing aerobic interval versus continuous training in coronary artery disease patients: 1-year data from the SAINTEX-CAD study[END_REF][START_REF] Prescott | Cardiac rehabilitation of elderly patients in eight rehabilitation units in western Europe: Outcome data from the EU-CaRE multi-centre observational study[END_REF] .

The CRF improvement observed in our CAD patients is valuable because of its previously proven prognostic value [START_REF] Harber | Impact of cardiorespiratory fitness on all-cause and disease-specific mortality: advances since 2009[END_REF] . The highest observed CRF improvement, after the CR stay and maintained one year thereafter, concerning patients with the lower basal CRF value, suggests a greater survival in these high-risk CAD patients [START_REF] Rosa | Cardiac rehabilitation after acute coronary syndrome: do all patients derive the same benefit ?[END_REF][START_REF] Martin | Cardiovascular fitness and mortality after contemporary cardiac rehabilitation[END_REF] .

Globally, the prognosis for CAD and diabetic patients is improved with the CR program [START_REF] Jimenez-Navarro | Benefits of cardiac rehabilitation on cardiovascular outcomes in patients with diabetes mellitus after percutaneous coronary intervention[END_REF] .

However, data are conflicting concerning the benefit level of physical training in these patients.

The gain value reported by other studies was less than, or similar [START_REF] St | Effects of cardiac rehabilitation in diabetic patients: both cardiac and noncardiac factors determine improvements in exercise capacity[END_REF][START_REF] Mourot | Cardiovascular rehabilitation in patients with diabetes[END_REF][START_REF] Nguyen | Factors associated with change in cardiovascular fitness for patients with peripheral and coronary artery disease in cardiac rehabilitation[END_REF] to our study. Improvement in CRF in active smoking CAD patients was also less than that in non-smokers. This is consistent with the deleterious consequences of active smoking on CRF [START_REF] Chen | Immediate effects of smoking on cardiorespiratory responses during dynamic exercise: arm vs. leg ergometry[END_REF] . Smokers must always be encouraged to undertake regular physical training although evidence of benefit on smoking cessation remains weak [START_REF] Ussher | Exercise interventions for smoking cessation[END_REF] .

A North-American study has reported a decrease in PA in a CAD population two months after their hospitalization [START_REF] Reid | Determinants of PA after hospitalization for coronary artery disease: the Tracking Exercise After Cardiac Hospitalization (TEACH) Study[END_REF] . Therapeutic patient education aims at teaching patients how to reinforce their objectives by acting on all risk factors and to increase the level of their PA [START_REF] Ambrosetti | Secondary prevention through comprehensive cardiovascular rehabilitation: From knowledge to implementation. 2020 update. A position paper from the Secondary Prevention and Rehabilitation Section of the European Association of Preventive Cardiology[END_REF] . Nevertheless, therapeutic education alone does not seem effective enough [START_REF] Anderson | Patient education in the management of coronary heart disease[END_REF] .

Data from PA self-questionnaires should be interpreted with caution because of the frequent overestimation reported, hence the choice in this study to measure the exercise capacity by a test [START_REF] Prince | Comparison of self-reported and objectively measured levels of sitting and PA and associations with markers of health in cardiac rehabilitation patients[END_REF] . Furthermore, many studies confirmed that the maintenance of a PA during phase III rehabilitation improved the CAD patient's survival rate [START_REF] Stewart | PA and mortality in patients with stable coronary heart disease[END_REF][START_REF] Ekblom | Increased PA post-myocardial infarction is related to reduced mortality: results from the SWEDEHEART register[END_REF] .

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Long term improvement of individual CRF is the main goal of the CR and should be considered as the major prognostic factor. This would be best achieved with an individually adapted and supervised physical training program, and associated with patient education provided by a multi-professional team. Moreover, we could speculate that the duration of the CR program should be adapted to the initial CRF value, with more physical training for patients with lower CRF values, and more education for patients with higher CRF values.

INTERESTS AND LIMITS OF THE STUDY

This study concerned the real-life typical cardiac rehabilitation. It could be reproduced in other countries presenting different rehabilitation modalities. Multicenter data also provided a real insight into the current situation.

Patients with severe heart failure or who had undergone recent non-coronary surgery were not included. Moreover, the results of this study concerned patients who agreed to participate in the protocol. The included population was relatively young and with a very small participation of females [START_REF] Rengo | Response to exercise training during cardiac rehabilitation differ by sex[END_REF] .

No control group was included, however spontaneously, non-rehabilitated CAD patients do not practice any PA after an ACS due to habits or fear [START_REF] Reid | Determinants of PA after hospitalization for coronary artery disease: the Tracking Exercise After Cardiac Hospitalization (TEACH) Study[END_REF][START_REF] Kronish | Objectively measured adherence to PA guidelines after acute coronary syndrome[END_REF] .

The CRF has been evaluated with the FRIEND formula. Despite the fact that cardiopulmonary exercise tests were not performed, patients with beta-blockers (90 to 95%) achieved the predicted maximal heart rate according to the Brawner formula for patients with beta-blockers (164 -0.7 x age) [START_REF] Brawner | Predicting maximum heart rate among patients with coronary heart disease receiving adrenergic blockade therapy[END_REF] .

During the follow-up period, the PA level evaluation had not been considered because of the difficulties encountered in estimating real PA. More precise evaluation with activity sensors would have been useful. In accordance with the data of the EU-CaRE study, the increase in ACCEPTED MANUSCRIPT / CLEAN COPY HDL-cholesterol that we observed was consistent with PA maintenance and reinforces the results of our study [START_REF] Prescott | Cardiac rehabilitation of elderly patients in eight rehabilitation units in western Europe: Outcome data from the EU-CaRE multi-centre observational study[END_REF][START_REF] Kelley | Aerobic exercise and lipids and lipoproteins in patients with cardiovascular disease: a meta-analysis of randomized controlled trials[END_REF] .

Our study did not require identical training protocols, only the number of sessions and their duration were collected. Thus, it was not possible to compare continuous endurance with interval training. However, the SAINTEX study did not find any difference between these modalities one year after the physical training period [START_REF] Rosa | Cardiac rehabilitation after acute coronary syndrome: do all patients derive the same benefit ?[END_REF] .

Lastly, 12-month follow-up does not predict the long-term effect of CR on maintenance of CRF and control of risk factors [START_REF] Berent | Lack of supervision after residential cardiac rehabilitation increases cardiovascular risk factors[END_REF] .

Our study did not include specific follow-up and could be used as reference for French CR middle term results. Thus, we could not evaluate the patient's leisure-time PA. This reflects the current reality, and our results could probably be improved with an adapted intervention (mobile messages, motivational interviews, psychosocial support ...), which are good topics for further studies [START_REF] Martinello | The Effectiveness of Interventions to Maintain Exercise and PA in Post-Cardiac Rehabilitation Populations. A systematic review and meta-analysis of randomized controlled trials[END_REF] .

CONCLUSION

In a French coronary patient cohort, among completers who agreed to enroll in this study, the improvement in functional capacity observed after a standard cardiac rehabilitation was maintained one year after the end of the stay for most coronary patients, without any specific management during the follow-up. 3.

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Figure1.

  Figure1. Flow diagram of study participants. ET indicates exercise test; CR, cardiac

Figure 2 .

 2 Figure 2. Evolution of the exercise tests (ET): the columns indicate the workloads at the first

Table 1

 1 Baseline data (n = 259)

	Demographic data
	Men		230 (89)
	Age, yr		60 ± 10
	Pathology
	ACS	STEMI	149 (58)
		NSTEMI	51 (20)
		Anterior	81 (31)
		Inferior	74 (29)
		Lateral	16 (6)
		Undetermined	29 (11)
	LVEF, %		55 ± 9
	Coronary angiography
		Single-vessel	122 (47)
		Two-vessel	67 (26)
		Three-vessel	70 (27)
	Coronary revascularization
		PCI	189 (73)
		CABG	72 (28)
		Complete	211 (81)
		None	5 (2)
	Data are expressed as number (%), or mean ± standard deviation
	Abbreviations: ACS, Acute Coronary Syndrome; CABG: Coronary Artery By-pass Graft;
	LVEF: Left Ventricular Ejection Fraction; PCI: Percutaneous Coronary Intervention;
	NSTEMI: Non-ST segment Elevation Myocardial Infarction; STEMI: ST segment Elevation
	Myocardial Infarction.

Table 2

 2 NA Data are expressed as number (%), or mean ± standard deviation Abbreviations: BMI, Body Mass Index; CHD, Coronary Heart Disease;CR, cardiac rehabilitation; NA, not adapted;

	Cardiovascular risk factors
		Beginning of CR	1-year post CR	P Value
		n=259	n=259
	BMI, kg/m 2	26.4 ± 3.7	26.7 ± 3.7	.06
	Waist size, cm	97 ± 11	97 ± 11	.19
	Smoking
	Never	100 (40)	100 (40)	.46
	Former	119 (48)	128 (51)
	Current	38 (15)	29 (12)
	High Blood Pressure	115 (44)	NA
	Diabetes	47 (18)	NA
	Hypercholesterolemia	153 (59)	NA
	LDL Cholesterol, mg/dL	85 ± 39	79 ± 28	.06
	HDL Cholesterol, mg/dL	44 ± 13	50 ± 14	< .001
	Family premature CHD	97 (37)

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Table 3

 3 Results of the exercise testing Data are expressed as number (%) or mean ± standard deviation Abbreviations: CRF, cardiorespiratory fitness; ET1, exercise test before training program; ET2, exercise test at the end of training program; ET3, exercise test one year after the end of training program; HR, Heart Rate; bpm, beat per minute; HRR, Heart Rate Reserve; MET, Metabolic Equivalent of the Task.

		ET1	ET2	P Value	ET3	P Value
				ET1 vs. ET2		ET2 vs.ET3
		n = 259	n = 259		n = 259
	Resting HR, bpm	66 ± 13	64 ± 10	<.008	65 ± 11	.22
	Peak HR, bpm	116 ± 19	125 ± 18	<.001	130 ± 19	<.001
	HRR, bpm	50 ± 19	61 ± 18	<.001	65 ± 19	<.001
	Workload, W	110 ± 37	139 ± 43	<.001	144 ± 46	<.001
	Relative workload, W/kg	1.41 ± 0.46	1.78 ± 0.53	<.001	1.83 ± 0.56	<.003
	Estimated CRF, MET	5.3 ± 1.4	6.4 ± 1.6	<.001	6.6 ± 1.7	<.002

ACKNOWLEDGEMENTS

We warmly thank the investigators of the following participating centers: Dr Sophie Lamoureux (Lyon), Dr. Julie Darchis (Machecoul), Dr Bénédicte Patois-Vergès (Dijon), Dr Lamia Tartière (Hyères), Dr Véronique Berder (Rennes), Dr Sonia Corone (Bligny), Dr. Bernard Pierre (Marcy l'Etoile), Dr. Anne-Lise Gugenheim (Corbie), Dr Véronique Gebuhrer (Villeurbanne), Dr Jean-Michel Guy (Saint-Priest en Jarez), Dr Nataliya Hrynchynshyn (Pontoise), Dr Eric Gobin (Noirmont) and Dr. Charlotte Mestrallet (Rumilly). Special thanks to Nathaniel Symes for his proof-reading.

FUNDING

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

AUTHOR CONTRIBUTION

BP, SK, BPV, SC, MCI, ACS and FC contributed to conception and design of the study. BP, SK, CR, EM, HK, AT contributed to acquisition of data. BP, MCI, ACS and FC drafted the ACCEPTED MANUSCRIPT / CLEAN COPY manuscript, SK, CR, EM, HK, AT, MCI, ACS and FC critically revised the manuscript. All authors have read and approved of the manuscript.

DECLARATION OF CONFLICTING INTERESTS

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.