Roberto Miorelli

Clement Fisher

Andrii Kulakovskyi

Bastien Chapuis

Olivier Mesnil

Oscar D'almeida

Defect sizing in guided wave imaging structural health monitoring using convolutional neural networks

Keywords: Structural health monitoring, inversion, deep neural network, convolutional neural network, defect sizing, guided wave imaging

This paper proposes an automatic defect localization and sizing procedure for Structural Health Monitoring based on guided waves imaging. The procedure is applied to an aluminum plate equipped with active piezoelectric sensors. The defect localization and sizing strategy is obtained through to the use of a convolutional neural network trained exclusively on numerical simulations of guided wave signals and post-processed by the delay and sum imaging algorithm. The paper shows the effectiveness of the proposed approach to invert both synthetic and experimental data.

Introduction

Structural Health Monitoring (SHM) identifies the ensemble of non destructive methods and techniques allowing to monitor the health of a structure under test throughout its lifetime with permanent sensors. The SHM paradigm is particularly adapted for inspecting a structure with specific constraints such as accessibility, for continuous diagnosis and prognosis tasks, or on-demand remote maintenance. SHM techniques allow to test very large portions of the structure by using a sparse network of sensors to automate inspections over large areas. The main industrial sectors interested by SHM are the energy and oil&gas industries with pipes monitoring, the aeronautic industry with the monitoring of large structures (fuselage, wings, etc.) as well as the the civil engineering industry.

In the context of SHM, a very promising inspection technique consists of considering ultrasound Guided Waves (GWs). Indeed, GWs can propagate in thin structures over long distances with little attenuation. Therefore, the measurement of interactions between specimen inhomogeneities (i.e. void, delamination, corrosion crack, hole etc.) located far away from the sensors is possible.

A typical ultrasound GW-SHM inspection scenario consists of a sparse set of piezoelectric sensors attached to the studied structure to generate and measure GWs signals. Most often, time domain GWs signals are post-processed in order to enhance the detection, localization and characterization of defects by employing qualitative imaging algorithms. Indeed, compared to analysing raw GW signals, working directly with images facilitate the human interpretation and make the analysis much easier. In the literature a broad set of GW Imaging (GWI) algorithms has been developed and studied [START_REF] Zhao | Active health monitoring of an aircraft wing with an embedded piezoelectric sensor/actuator network: Ii. wireless approaches[END_REF][START_REF] Michaels | Detection, localization and characterization of damage in plates with an in situ array of spatially distributed ultrasonic sensors[END_REF][START_REF] Quaegebeur | Correlation-based imaging technique for fatigue monitoring of riveted lap-joint structure[END_REF][START_REF] Hall | Guided wave damage characterization via minimum variance imaging with a distributed array of ultrasonic sensors[END_REF][START_REF] Zou | Highly accurate online characterisation of cracks in plate-like structures[END_REF].

In this paper, the Delay-And-Sum (DAS) [START_REF] Michaels | Detection, localization and characterization of damage in plates with an in situ array of spatially distributed ultrasonic sensors[END_REF] GWI algorithm is used to generate the images associated to the signals measured by the piezoelectric sensor network. The biggest merit of the DAS in the context of this paper is the fact that it provides easily interpreted images with low computational burden, and only requires a limited knowledge on the GW propagation within the structure.

A major challenge in GW-SHM is to automate the diagnostic operations in a reliable fashion.

Among the different objectives are automated detection (e.g., healthy vs. damaged structure), classification (e.g., defect(s) type classification) and regressions tasks (e.g., defect(s) sizing). Machine Learning (ML) algorithms such as kernel machines [START_REF] Scholkopf | Learning with kernels: support vector machines, regularization, optimization, and beyond[END_REF] and Deep Neural Network (DNN) [START_REF] Goodfellow | Deep learning[END_REF] have spread recently in various industries to solve tasks such as security and video surveillance, image recognition, natural language interpretation, etc.. One of the main advantages of using DNN compared to kernel machines algorithms, consists of the fact that the feature engineering stage can be avoided, which is not the case with kernel-based techniques, largely limiting the deployment of the latter to a narrow spectrum of use cases due to the need of human intervention. Indeed, loosely speaking, a DNN architecture extracts automatically the most suitable features through the optimization of the architecture's millions or even billions of weights. The drawbacks of such an approach is first that DNN are often seen as black-box objects of which the inner operations are difficult to understand and second that it requires relatively large databases for training.

In the framework of Non Destructive Testing (NDT), the use of ML methods has been proposed with good success in eddy current testing [START_REF] Ahmed | Real time groove characterization combining partial least squares and svr strategies: application to eddy current testing[END_REF][START_REF] Ahmed | An adaptive sampling strategy for quasi real time crack characterization on eddy current testing signals[END_REF][START_REF] Salucci | A nonlinear kernel-based adaptive learning-by-examples method for robust ndt/nde of conductive tubes[END_REF][START_REF] Zhu | A novel machine learning model for eddy current testing with uncertainty[END_REF], infrared thermography [START_REF] Lu | Automatic relevance determination of adaptive variational bayes sparse decomposition for micro-cracks detection in thermal sensing[END_REF][START_REF] Luo | Temporal and spatial deep learning network for infrared thermal defect detection[END_REF][START_REF] Duan | Automated defect classification in infrared thermography based on a neural network[END_REF][START_REF] Hu | Lstm-rnn-based defect classification in honeycomb structures using infrared thermography[END_REF], ultrasound testing [START_REF] Ye | Computerized ultrasonic imaging inspection: From shallow to deep learning[END_REF][START_REF] Almansouri | Deep neural networks for non-linear model-based ultrasound reconstruction[END_REF][START_REF] Munir | Convolutional neural network for ultrasonic weldment flaw classification in noisy conditions[END_REF][START_REF] Posilović | Flaw detection from ultrasonic images using yolo and ssd[END_REF], or even X-ray [START_REF] Lin | Detection of a casting defect tracked by deep convolution neural network[END_REF][START_REF] Zhu | The defect detection algorithm for tire x-ray images based on deep learning[END_REF][START_REF] Mery | Aluminum casting inspection using deep learning: A method based on convolutional neural networks[END_REF]. Even though the various NDT inspection methods output data of different nature, a ML process can be tailored for each application.

Typically, eddy current testing deals with complex valued measurements, X-ray data are represented by gray scale images whereas infrared thermography data is described by real valued time domain signal. The latter is also often used to represent ultrasound signals (i.e., A-and B-scans).

The various format of the input data of the ML steps requires a specific care on the selection and tailoring of the specific ML procedure to be used in order to reach satisfying performance. For example, the cardinality (i.e. size of the inputs) of the problem may greatly impact the overall performance in terms of accuracy and efficiency. In this context, the objective of this paper is to develop a DNN architecture to automatically invert defect position and size based on GWI images generated by simulations. The motivation to use simulations to train the DNN is to limit the cost of database generation, in general much lower in simulation than in experiment. Such approach however requires a thoroughly validated simulation tool within the perimeter of parameters used for the training. The main contributions of this paper are the following.

1. In contrast to many other works in the literature such as [START_REF] Ye | Computerized ultrasonic imaging inspection: From shallow to deep learning[END_REF][START_REF] Almansouri | Deep neural networks for non-linear model-based ultrasound reconstruction[END_REF][START_REF] Munir | Convolutional neural network for ultrasonic weldment flaw classification in noisy conditions[END_REF][START_REF] Posilović | Flaw detection from ultrasonic images using yolo and ssd[END_REF] addressing classification tasks based on ultrasound testing, the regression problem is solved automatically to invert defect size and location. Toward this end, a DNN approach exploiting a Convolutional Neural Network (CNN) architecture specifically developed to deal with GWI data is proposed. For the sake of brevity, this network architecture is referred to as ShmGwi-InvNet.

2. Within the framework of GWI data as CNN input, the interpretability of the per-layer operations of the CNN is discussed in order to shed some light on the inner operations of the CNN architecture under consideration. That is, considering the response of the neurons at different depths (i.e., the feature maps), the pertinence of the features extracted during the training phase are analysed. These indications can be used to interpret and explain the working mechanism associated to the designed network, hence provide input to improve the network architecture and mitigate the "black-box effect".

3. A third distinguishing point of this contribution resides on the use of synthetic data needed to generate the set of training samples to feed the ShmGwi-InvNet. The inversion performance has been tested on both synthetic (i.e. simulated) and experimental GWI data even though the training is done exclusively on simulated data. This choice requires to develop a robust calibration procedure between simulated and experiment inputs. This last step is especially critical because it demonstrates that a numerically trained neural network can be directly used to invert experimental data and, because large database generation is easier to conduct numerically than experimentally, it greatly widens the potential field of application of such techniques.

The GW-simulations used in this paper are achieved using the SHM module of the CIVA software, which relies on a high order finite element technique [START_REF] Mesnil | Validation of spectral finite element simulation tools dedicated to guided wave based structure health monitoring[END_REF] to ensure fast computations without simplifying geometries or requiring dedicated computing hardware. Because the simulations aspects are of little importance for this paper, the reader is invited to refer to [START_REF] Imperiale | A macro-element strategy based upon spectral finite elements and mortar elements for transient wave propagation modeling. application to ultrasonic testing of laminate composite materials[END_REF] for more details.

Other simulation processes could indeed be used as long as two requirements are met: first the match between simulations and experiments must be sufficient and second, simulations must be fast enough to ensure large database generations. Other relevant simulation techniques include boundary element method [START_REF] Li | Boundary element modelling of ultrasonic lamb waves for structural health monitoring[END_REF] or GPU-accelerated finite elements [START_REF] Shen | Hybrid local fem/global lisa modeling of damped guided wave propagation in complex composite structures[END_REF].

The paper is structured as follows: In Section 2 the studied configuration and the imaging process are presented, along with the calibration procedure. In Section 3 the ShmGwi-InvNet architecture specifically developed for GWI inversion is presented, and is then used to obtain the synthetic and experimental results analyzed and discussed in Section 4. In the latter section, some insights on the inner working mechanism associated to DNN architecture developed through the study of features extracted at different layers are also provided. Conclusions, remarks and the ongoing research axes on the application of ML to automatic detection and regression applied to SHM problems are provided in the last section of this paper.

SHM guided waves imaging

This section describes the GWI technique employed to post-process ultrasound GWs signals measured by a piezoelectric sensor network attached to the structure of interest. The outcome of the GWI process is image-like signals containing features that can be more easily interpreted by human operators than GWs raw signals in the time domain. Indeed, being multimodal and dispersive, GW raw signals are too complex to be analysed directly by an human operator. Moreover, if P stands for the number of piezoelectric sensors probing the structure, the number of GWs signals to be analysed is equal to P × (P -1) raw signals if pulse-echo signals are not recorded. Hence the GWI is an excellent way to combine all the measurements into a single easily interpretable image.

The following section provides a brief overview on the delay-and-sum GWI algorithm [START_REF] Michaels | Detection, localization and characterization of damage in plates with an in situ array of spatially distributed ultrasonic sensors[END_REF] applied to SHM problems. It is worth mentioning that other kinds of GWIs algorithms such as Rapid, Excitelet or Minimum Variance [START_REF] Zhao | Active health monitoring of an aircraft wing with an embedded piezoelectric sensor/actuator network: Ii. wireless approaches[END_REF][START_REF] Quaegebeur | Correlation-based imaging technique for fatigue monitoring of riveted lap-joint structure[END_REF][START_REF] Kulakovskyi | Defect imaging on cfrp and honeycomb composite structures by guided waves generated and detected by a sparse pzt array[END_REF][START_REF] Hall | Guided wave damage characterization via minimum variance imaging with a distributed array of ultrasonic sensors[END_REF] can also be exploited for inversion following the procedure detailed in this paper. However in this work, the DAS algorithm has been chosen since it provides a good trade-off between robustness, implementation simplicity and computational efficiency.

The delay and sum algorithm applied to SHM

The SHM configuration under study is shown in Figure 1. It is an aluminum panel instrumented with 8 circular piezoelectric transducers. The DAS algorithm can be formulated as follows: Let us consider an SHM system composed of P piezoelectric sensors with a round-robin scanning procedure, i.e. each transducer sequentially generates a ultrasonic pulse while the other P -1 remaining sensors measure the propagated wavepackets. Subsequently, a discrete 2D-grid is defined over the whole area of interest. An arbitrary grid point is identified by its coordinates (x, y) to which is associated a Damage Index (DI) given as DI (x, y) =

P (P -1) k=1 r k τ k (x, y) (1)
where r k (•) is the envelope of the residual signal of the k-pair of sensors, defined as the difference between the signal in the pristine state and the signal in the damage state, and τ k (x, y) represents the GWs theoretical time of flight associated to the inspection point (x, y) for the transmission receiver path numbered k.The theoretical time of flight τ k (x, y) is computed assuming no dispersion by :

τ k (x, y) = τ (x, x i , x j ; y, y i , y j ) = 1 Cg (x -x i ) 2 + (y -y i ) 2 + (x -x j ) 2 + (y -y j ) 2 (2)
In Eq. 2, the subscripts i and j identify the positions associated to the emitting and receiving PZT sensors corresponding to the path of index k, respectively. C g stands for the group velocity associate to the path going from the i-th transmitter to the j-th receiver passing through the inspection point located at (x, y). Typically such a value can be either inferred by simulations or measured. Note that in DAS, C g is constant, i.e. the dispersion effect are neglected, which is a reasonable assumption as long as the excitation pulse is narrow-band. Note that for anisotropic propagation media such as composites, C g is a function of the propagation direction. The envelope of the residual signal in [START_REF] Zhao | Active health monitoring of an aircraft wing with an embedded piezoelectric sensor/actuator network: Ii. wireless approaches[END_REF], is computed as

r(t) = s r (t) 2 + H (s k r (t)) 2 where H s k r (t) 2 refers
to the Hilbert transform and

s r (t) = s c (t) -s b (t) (3)
is the residual signal calculated by subtracting the current state signal s c (t) from the pristine baseline signal s b (t). Note that the DAS process is similar to the SAFT (Synthethic Aperture Focussing Technique) [START_REF] Sicard | Guided lamb waves and l-saft processing technique for enhanced detection and imaging of corrosion defects in plates with small depth-to wavelength ratio[END_REF] with the following two differences. First, in DAS the baseline signals are used (Eq 3) to better highlight the signature of the defect and be more robust to additional effects, such as edge reflections. Second, in SAFT, a unique line of sensors is often used to image the structure, assumed to be unidirectional in the direction orthogonal to the sensor line while in DAS the monitored structure is often surrounded by sensors to ensure multidirectional inspection and have a better chance to monitor directive defects such as cracks.

The application of the DAS imaging process is illustrated in Fig. 2 the damage is highlighted in red, the subsequent wavepackets being the reflections on the edges of the plate. Note that a single signal path is represented here while a total of P × (P -1) = 56 such signals are used for imaging. In Fig. 2b is shown the DAS image computed by the previously described process. In this result, a high pixel intensity is represented by the blue color and highlights a high damage index. Straightforward interpretation of GWI can be a difficult task and and human-error are possible due to the presence of false alarms (as can be seen in Fig. 2b) on the zone close to the central-right sensor). The use of the machine learning strategy presented in this paper aims to mitigate this issue.

Model-driven supervised learning based on GWI data

In this section, the details about the model-driven methodology developed in this paper to design the CNN architecture for defect sizing based on DAS images are presented. First some details about the generation of the datasets required for training, validation and testing are provided.

The datasets are based on synthetic GWI images generated through the CIVA software [START_REF] Mesnil | Validation of spectral finite element simulation tools dedicated to guided wave based structure health monitoring[END_REF][START_REF] Imperiale | A macro-element strategy based upon spectral finite elements and mortar elements for transient wave propagation modeling. application to ultrasonic testing of laminate composite materials[END_REF].

Subsequently, the CNN architecture is developed, detailing the various layers.

Deep learning inversion strategy based on convolutional neural networks

In NDT and SHM, the availability of experimental labelled data may be limited and does not allow to effectively train machine learning algorithms. Note that for SHM, the limited data availability problem might be solved for specific use-cases in which sensor deployment and data collection over time has been achieved, but the collected data will remain use-case specific and are unlikely to be directly useful for other applications. To overcome this issue, synthetic data is generated by the SHM module of the CIVA software [START_REF] Mesnil | Validation of spectral finite element simulation tools dedicated to guided wave based structure health monitoring[END_REF]. The software provides the possibility to simulate GWs signals and subsequently post-process them by using the DAS algorithm.

A dataset is defined as a collection of input-output couples as

D = {(X 1 , y 1 ) , ..., (X N , y N )} ⊆ (X×Y) N , (4)
with N stands for the number of samples contained in the dataset. X i is a matrix containing the i-th GWI image such that X i ∈ R N x×N y with N x and N y representing the number of pixels used to discretize the image along the x and y directions, respectively. y i ∈ R Np is the vector of N p entries associated to the flaw parameters used to simulate the i-th DAS image, i.e. the flaw size and position. In other words, the supervised learning strategy employed in this paper aims at mapping the space of the GW post-processed signals (i.e., the features represented by the GWI) to the space of flaws parameters. This approach is schematized in Fig. 3. In the next section, the details about the CNN architecture developed to effectively carry out this mapping are presented.

The ShmGwi-InvNet architecture

In the following, more insights about the CNN employed to map the GWI data on the model parameter space block labelled "ShmGwi-InvNet" illustrated in Fig. 3 are provided. In this work, due to the cardinality of the addressed problem (i.e., each image has N x ×N y pixels), a deep learning approach based on multi-layer perceptron (MLP) easily turns to be a computationally demanding task. Indeed, a typical GWI image is made of around N x × N y = 100 × 100 = 10 4 pixels. As a consequence, the number of hidden units (i.e., the neurons) and the number of hidden layers cannot be increased without a heavy impact on the number of parameters to be estimated [START_REF] Safran | Depth-width tradeoffs in approximating natural functions with neural networks[END_REF]. Besides the obvious increase in terms of computational time, the second drawback consists of the fact that having more parameters to tune requires more training data to avoid overfitting. Thus, the training set generation turns to be very costly and computationally expensive. A widely employed approach to tackle this problem consists of adopting a CNN-based deep learning paradigm. Compared to MLP, CNN architectures enable sparse interaction by employing convolutional kernels, sharing the weights trough different layers hence limiting the number of parameters to be estimated [START_REF] Goodfellow | Deep learning[END_REF].

Furthermore, CNN naturally allow to work with 2D-images without having to go through the vectorization of matrices that may break down the spatial coherence of images leading, for certain cases, to a more difficult learning procedure.

The ShmGwi-InvNet network architecture is represented in Fig. 4a) and can be described as follows. The ShmGwi-InvNet is composed of an input layer of size N x × N y , i.e. the size of the GWIs, followed by three convolutional layers, two fully-connected MLP layers and an output layer. Each of the convolutional blocks is composed of the concatenation of elementary tasks performing convolution, activation function, and max-pooling operations as sketched in Fig. 4b).

x ×N

(1) y = N x ×N y ) and applies on convolutional F (k) filters with size F (k)

x , F Thus the obtained feature maps are vectorized and connected to two layers of fully-connected neural network made of 128 and 32 neurons, respectively, and activated by the ReLU function.

The last fully connected layer is the output layer composed of N p output neurons. The weights optimization has been carried out by using Adam optimization algorithm. The architecture was implemented by using the Python library Tensorflow [START_REF] Abadi | Tensorflow: Large-scale machine learning on heterogeneous distributed systems[END_REF].

Numerical and experimental validation

In order to assess the performance of the ShmGwi-InvNet numerical and experimental vali- then directly comparable, the difference between the two being limited to the phenomena not described in the simulations and assumed to be negligible (measurement noise, error in the finite element model convergence, attenuation etc.). For the subsequent approach to work, the match between experimental and calibrated simulation signals must be excellent. This assumption has been verified by the authors in previous work in a very similar configuration [START_REF] Mesnil | Validation of spectral finite element simulation tools dedicated to guided wave based structure health monitoring[END_REF].

ShmGwi-InvNet inversion results

In this section, regression results aiming at retrieving defect positions and defect sizes from both synthetic and experimental tests sets are presented. As a reminder, the CNN is exclusively trained with the calibrated numerical DAS images, then used to invert both numerical and experimental data, without any modification. In Fig. 6, predictions obtained by the ShmGwi-InvNet are provided in terms of true versus predicted values, respectively represented by blue circles and red crosses for the simulated and experimental data. The predictions on experimental data are overlapped to the simulated one to compare the variance of the inversion. As one can notice, in the case of simulated data, the scattered points are aligned on the diagonal solid line that indicates the best possible agreement between predictions and ground truth. The predictions associated to the experimental data show that the estimation of defect positions is slightly more spread than the synthetic counterpart. Furthermore, defect radius characterization based on experimental data is successful as well and is as spread as the simulated points. In other word, no significant error is induced in the inversion of experimental data compared to the inversion of simulated data.

Concerning the experimental test set, it is worth mentioning that the predictions with the two largest errors are the ones associated to values on the training set boundaries (i.e. defect radius equals to 2.5 mm and 7.5 mm). This values are most "critical" due to the absence of a sufficient number of proxy training samples due to the chosen parameters' variation range.

Quantitatively, the accuracy of the predictions can be quantified by using the Mean Absolute

Error M AE = T i=1 |y i -ŷi | /T , the Root Mean Squared Error RM SE = T i=1 (y i -ŷi ) 2 /T and through the correlation coefficient defined as R 2 = 1-T i=1 (y i -ŷi ) 2 / T i=1 (y i -y) 2 . Where, for each defect parameter, y i , ŷi and y represents the i-th actual parameter, the predicted one and the mean value of the test set, respectively and T is the test set size. In Tab. 1 the computed errors based on the entire testing set are provided. In Fig. 7, the prediction error analysis are represented in terms of kernel density estimation computed for each parameter at different defect radius. The analysis of these plots shows that the two localization parameters are almost independent of the flaw radius and the absolute error lesser than 10 mm regardless of whether the horizontal and vertical positions is considered. Concerning the radius estimation, the absolute error is less than 0.5 mm and is completely independent on the flaw radius size. Compared to the A0 interrogating wavelength at 40 kHz of 25 mm, these performance can be judged as excellent.

From a computational standpoint, the ShmGwi-InvNet has been trained in 1200 epochs achieved spatial location associated to the defect. Furthermore, looking at the feature maps for both the considered images, one can notice that the most active neurons for the feature maps in Fig. 8a) are the less active in Fig. 8b). Such information shows how the CNN has learnt during the training phase by optimizing the network weights.

Conclusion and perspectives

In this paper, an inversion strategy based on a deep convolutional neural network is presented.

The approach relies on a model-driven inversion where simulations conducted with CIVA have been used to train the CNN model. The performance of the proposed CNN architecture are assessed on both simulated and experimental data after training on exclusively simulated data.

On both the datasets, successful inversion of defect position and size is conducted, with an average localization error of 10 mm and a sizing error or 0.5 mm, with an interrogating wavelength of 25 mm. Moreover, the experimental predictions present good correlation with the simulated one.

This agreement shows that the model-driven approach can be successfully employed to train a robust ML inversion model against unseen experimental and synthetic samples based on GWIs, and quantify features with an accuracy far below the wavelength. Note that this conclusion hold only as long as the simulated data matches reasonably well the experimental data. Furthermore, some insights on the interpretation of the features extracted by the CNN and its inner working mechanisms were presented.

The possible extensions of this works are multiple. The main topic of interest is studying and possibly enhance the capability of the ShmGwi-InvNet architecture on more challenging problems such as the case of multiple-defects problem or to address other kind of defect shape and/or topology. Moreover, the extension and the validation of the proposed model-driven approach to more challenging propagation phenomena as the GW signal in composite materials is also targeted.

On the long term, such approach could be very useful to conduct an automatic diagnostic of GW signals on a structure in which experiments cannot be easily conducted (either for cost or practical reasons). To reach such applications however, exhaustive validation of models would be required to ensure that simulations provide realistic results, including in varying environments.

Figure 1 :

 1 Figure 1: SHM experimental setup with in a) the aluminium plate with eight piezoelectric sensors encircling area to monitor damaged by a through-hole and b) a sketch of the experimental measurement procedure

  : first, an example of pristine (blue) and damaged (orange) signals are shown at the top of Fig 2a. The corresponding residual signal (i.e. the difference) is shown just below. For clarity purposes, the incident wave-packet from

Figure 2 :

 2 Figure 2: An example of experimental GWs signals leading to a DAS image with a) raw signal (top) and residual signal (bottom) for an emitting-receiving couple with a 40kHz 2-cycle tone burst and b) The resulting DAS image; the piezoelectric sensors are represented by the black dots and the colormap is the Damage Index.

Figure 3 :

 3 Figure 3: Proposed model-driven inversion scheme based on CIVA.

Figure 4 :

 4 Figure 4: The ShmGwi-InvNet architecture is shown in a) from input to output layers. In b) a unitary convolutional block architecture is represented.

  and a stride equal to one. In the activation layers, the computed filter responses passes through the Rectified Linear Unit (ReLU) activation function f ReLU (z) = max(0, z). Subsequently, the output passes through the pooling layer operating independently on each of the activation maps by performing a maxpooling operation based on a filter of size P reducing the image size by a factor of four. These operations are repeated two times, where 64 and 128 feature maps have been chosen for the first and the second convolutional block, respectively.

4 . 1 .Figure 5 :

 415 Figure 5: Normalized GWI obtained via DAS post processing. In a) and c), the experimental maps for two defects having radius equal to 4.5 mm and 6.0 mm, respectively. In b) and d), the simulated maps associated to the experimental test samples in a) and c), respectively .

Figure 6 :

 6 Figure 6: ShmGwi-InvNet regression results obtained for the following parameter of the defect: a) x-position, b) yposition and c) radius where "" indicates a prediction based on synthetic data whereas "×" stands for a prediction based on experimental data.

Figure 7 :Figure 8 :

 78 Quadro M1200 with 4 GB of RAM.

Table 1 :

 1 Prediction errors associated to the synthetic and experimental test sets.

		M AE [mm] RM SE [mm]	R 2 [a.u.]
		Sim. Exp. Sim. Exp.	Sim. Exp.
	x-pos. (p x )	3.72 21.53 4.43	9.9	0.14 0.36
	y-pos. (p y )	5.03 23.67 5.85 11.21	0.18 0.45
	Hole rad. (r) 0.98	-	0.98	-	0.98 0.91

Preprint submitted to NDT&E International December 1, 2023


Nathalie Laforge

Léa Wagner

Matthieu Jules

Anne-Gaëlle Planson

Construction of synthetic bacterial consortia based on metabolic mutualism

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

. The model assumes that as the growth of each strain is dependent on its partner, the growth rate of each subpopulation is identical. This assumption will be challenged with FACS analysis (on-going experiments).

Conclusion and perspectives

D

Within natural ecosystems, microorganisms live in communities and interact with each other1 . Division of Labor (DoL) is an evolutionary strategy widely used in these communities and allows for the distribution of metabolic tasks between individuals for mutual benefit. This strategy reduces the metabolic burden of each individual 2 and microbial communities exhibit selective advantages over monocultures in terms of stability, robustness, and functionality 1 . However, piloting population dynamics of cocultures remains challenging as the faster-growing species may dominate over time 3 . One possible strategy to maintain the coexistence of populations is cross-feeding of metabolites, a common process in natural consortia 4 .

This project aims to build mono-species synthetic consortia composed of Bacillus subtilis (Bsu) strains exhibiting obligate mutualism based on amino acids cross-feeding ( Viable Bacillus subtilis co-dependant consortia based on amino acids cross-feeding were identified via the screening of 36 strains in pairwise cocultures. The three consortia selected for further characterization, {ΔmetE, ΔargG}, {ΔpheA, ΔleuA} and {ΔlysA, ΔthrC}, are stable and their growth rate reaches 25 to 47% compared to the WT. Proportion of both partners evolve towards a different balance for each consortium.

The fitness of the selected consortia will be improved by directed evolution by combining two methodologies. An inducible hypermutator system, developed by Irène Tanneur (SyBER), will be integrated in the consortia. This system increases the mutation rate per generation by a factor of 10 to 1000. Such modified consortia will be cultivated in a continuous culturing device (Fig 5) developed by Cyprien Guérin and Pierre Nicolas (MaIAGE). This chemostat allows to continuously monitor the biomass, and automatically adjust the dilution rate to maintain a desired growth rate. The variants obtained from these experiments will be isolated and characterized in order to find mutations that optimize division of labor.

Fig 2 -

 2 Fig 2 -(A) A first pre-culture is performed in rich medium (LB), a second in minimal medium supplemented with glucose and malate (M9GM) and amino acids. The consortia are then cultivated in 384-well plates in M9GM. The OD 600nm is measured in a plate reader. (B) Growth curves of a consortium (in blue) and of the corresponding monocultures (in orange), compared to the WT (in grey) in 384-well plates. (C) Criteria for the comparison of all consortia. (D) Heatmap of the OD 600nm after 19h of culture. Each square is the mean of a duplicate. Color intensity indicates the growth percentage compared to the WT strain in monoculture. Selected consortia for further characterizations are outlined in yellow.

Fig 4 -

 4 Fig 4 -A logistic function (in red) is fitted to the experimental data (in orange). The end-point proportion of subpopulations allowed to determine k ΔlysA =0,16*k and k Δthrc =0,84*k, as well as to simulate growth of each partner (in blue).

Fig 1 -

 1 Fig 1 -Synthetic microbial consortium consisting of two B.subtilis strains. The exchange of amino acids, i.e., crossfeeding, is required for the growth of both strains.

Fig 5 -

 5 Fig 5 -Continuous culturing device developed by Cyprien Guérin and Pierre Nicolas (MaIAGE)



  Identification of viable B. subtilis mutualistic consortia.  Selection of {ΔmetE, ΔargG}, {ΔpheA, ΔleuA} and {ΔlysA, The selected consortia are stable over time. The selected consortia reach a growth rate of 0.18 h -1 to 0.33 h -1 .  Proportion of subpopulations reaches an equilibrium, {6% ΔmetE, 94% ΔargG}, {37% ΔpheA, 67% ΔleuA} and {16% ΔlysA, 84% ΔthrC}.

Fig

  t=19h

Rafieenia et al.

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