dataesr/scientific-paragraphs-categorization huggingface dataset:
[36]. However, to the best of our knowledge and up to now, no publication claims to have implemented it in CodeV. Moreover, the results obtained with a MSE-based criterion and a commercial lens design software have never been compared to other co-design methods. Indeed, there are several methods to optimize a hybrid system from end-to-end. In particular, another joint optical-digital design method has been developed by Burcklen et al. [1]  to facilitate the use of the co-design paradigm by professional optical designers in CodeV. It is based on a "surrogate" merit function that contains terms classically available in lens design software but used in a non-standard way to implicitly take into account image processing. Our purpose in this work is to implement the MSE-based optimization criterion in the CodeV software, and to compare the obtained performance with two other optimization criteria.
The example of application chosen for illustration is the design of a Cooke triplet aimed at providing good image quality over a wide field of view (FoV) of 40 • (±20 • ). The basic elements of the considered hybrid optical-digital system are described in Sec. 2. In Sec. 3, we apply the "conventional" design method, which considers the existence of the digital processing only after the optimization of the optical system. We establish the performance of the obtained hybrid system by evaluating its modulation transfer functions (MTFs), its effective MTF (taking into account the whole optical system and post-processing) and the visual quality of the restored images. In Sec. 4 , we investigate in the same way the method based on a surrogate criterion implemented in Code V  [1]. Then, in Sec. 5, we describe the implementation of a method for directly optimizing the MSE under CodeV, and compare the performance of the obtained hybrid system with those provided by the two other methods.
This can be done by adding penalization terms like p 2 × (f ′ -f ′ target ) 2 (where p is a positive constant, defining the "weight" of this term) to the basic criterion, or by strictly enforcing the constraint using Lagrange multiplier. In this study, we optimize the sum of the squares of the RMS diameters of the spot diagrams under common CodeV optical constraints (see Supplemental 1, section 1). The CodeV optical design software uses by default the Levenberg-Marquardt (or Damped least squares) algorithm to minimize this cost function  [16, 18, 9, 20] , but other algorithms can be used or implemented  [22, 15, 24]. Since the Levenberg-Marquardt algorithm is a local minimization algorithm, convergence to a global minimum is not guaranteed. The expertise of a lens designer is therefore necessary, both to choose a suitable starting point for the optimization  [6]  and to steer it towards an acceptable local minimum.
Analysis of final image quality Let us now verify the observations made on MTFs and effective MTFs through image simulations. Figure  3  shows the scene used for these simulations. It is a large image, of 5144 × 5144 pixels, chosen to obtain a FoV of 40 • (±20 • ) on the diagonal of the sensor, with a pixel size close to 5 µm. Several subparts are selected on the raw image, to check the reconstruction of the details at three different positions in the FoV: on axis, intermediary position and extremity. Simulation of the observation of this scene through the Cooke triplet is performed with the image simulation tool of the CodeV software, with 20 × 20 variable PSFs in the FoV, in order to take into account all the defects of the optical system. The image is then corrupted with a white Gaussian noise so that the SNR is 34 dB.
This approach will be called "co-optimization by MTFs equalization". It should be noted that the maximization of the MTF values is a very important point of this optimization technique. In addition to corresponding well to the goal of obtaining the best possible image quality over the whole FoV, this co-design criterion can be seen as an alternative (a "surrogate") to the MSE. Indeed, several studies have shown that optimizing performance in terms of MSE over a set of parameters (e. g. depths of field) leads to MTFs that are close to each other  [11, 1, 13]. This criterion can be implemented in various ways in commercial optical software  [1]. In this work, we have disabled the default CodeV error function based on the sum of squared spot diagram RMS diameters and only enforced the following user-defined constraints: • Favoring equality between the MTFs at a set of some Y-meridional FoV positions and the on-axis MTF (it is a set of weighted constraints, to allow some latitude and balancing on these equalizations).
Implementation and use of the MSE criterion So far, we have used the MSE to evaluate the performance of hybrid systems (through the IQ criterion), but not as a direct co-optimization criterion. An implementation of this criterion has already been done under Zemax  [30, 36, 35] , but until recently, it seemed to us impossible to do the same under the other well-known CodeV commercial lens design software, since it was not possible to directly access PSFs (or 2D OTFs) in a user-defined error function during optimization. Using the new features of CodeV version ≥ 11. 0 (2017) that remove these limitations, we have been able to implement -for the first time to our knowledge -MSE optimization directly into CodeV. The details of the implementation are given in Supplemental 1 (section 3). In this section, we cooptimize a Cooke triplet with this method and compare its performance with the two previously described triplets.
Conclusion In this article, we have compared three different methods to co-optimize a hybrid optical/digital imaging system with commercial lens design software: conventional optimization based on minimization of the RMS diameter of spot diagrams, a surrogate criterion based on near equality of the MTFs and a true MSE criterion taking explicitly into account the digital processing in the optimization process. To implement the latter method, we integrated -for the first time to our knowledge -MSE optimization to the CodeV software. These three methods have been illustrated and compared on a concrete application: the design of a Cooke triplet having good image quality everywhere in the FoV. The obtained results demonstrate the superiority of MSE co-optimization over the other methods, both in terms of quantitative and visual image quality.
A. 3 Implementation of MSE-based co-optimization in CodeV CodeV lens design software offers the ability to write advanced optimization macros, that may use externally coded efficient compiled extension in the form of Dynamic Link Libraries (DLLs). It also offers, since version 11. 0, the possibility to calculate the point spread function (PSF) of a given field during the iterations of its internal optimization algorithm, thanks to the predefined CodeV Macro-PLUS function PSF DATA 1FLD. Therefore, we create a DLL, based on a code written in C, taking as arguments the inputs (2D arrays describing the PSFs) and the output (the MSE) which is calculated according to the values of the PSFs. Note that computing a too large number of PSFs during an optimization routine would not be practicable due to large computing time.
For optimization purposes, these two defects will be considered negligible or easily corrected by post-processing, as they generally have little effect on the visual quality of an image. The written C code must be compiled as a 32-bits DLL (CodeV is only available for this software architecture) using a Synopsyssupplied Makefile, extended to allow FFTW use. This allows one to obtain a DLL (and associated files) that can easily be called in a homemade CodeV macro function. This function can then be called directly in the CodeV optimization routine, like other functions of interest to the optical designer. The optimization goal is then to make the obtained MSE result as close as possible to 0.
The training depends on the right-hand side and solution pair, thus results in a very large dataset, containing around 16. 2 million samples. As the proposed Green's function is still an approximation of the ground-truth, the authors adopted it as a pre-conditioner for the linear system. On the contrary, we adopt the approximated Green's function as a multi-level iterative solver. Green's functions were also explored in a multi-scale fashion to solve specific electromagnetic potentials. In a series of papers, Li et al. [5, 6, 7]  proposed neural networks as non-linear neural operators to map coefficients of differential operators directly to solutions. However, the proposed neural operators always assume fixed forcing terms in the PDE, and these methods do not deal with complex boundary conditions.
Predicting Green's Function for Various Boundary Geometries We further evaluate the performance of MLPs to predict Green's function kernels based on general boundary conditions of arbitrary scenes. Note that the training does not rely on either the solution or the right-hand-side vectors. It only requires SDF values for all levels and the corresponding discrete Laplacian operator (ϕ 1 ,. , ϕ L , A). Dataset generation. We randomly place spheres and rectangles of random sizes into the scene to represent interior Neumann boundaries. Spheres and rectangles keep canonical orientations for simplicity, but are allowed to overlap in order to create more complex shapes. SDF values are computed at different discretization levels from the parameters of the spheres and rectangles to obtain ϕ 1 ,. , ϕ L.
The voxelized obstacle setting and its corresponding discrete Laplacian A are computed   at the finest discretization level. The discrete Laplacian is not stored for coarser levels, instead, they are downsampled relative to the finest resolution on the fly during training. We generate 1,000 scenes for the dataset (see examples in Figure  6 ), but the number of training samples is larger as we randomly select local patches. Moreover, 10 scenes are generated in a similar fashion (randomly placing spheres and rectangles) for creating the testing dataset. This means that none of the test sampeles is seen during training. The left side of the exterior boundary of all scenes is assumed to be Dirichlet while the other sides are assumed to be Neumann. Predicting on new boundary settings. We train MLPs for grids with 129 × 129 nodes and show test results evaluated by solving Poisson equations in Figure  7 (a, b ).
Limitations Inaccurate MLP kernel predictions. As shown in Figure  7  (c), our trained model underperforms on some individual patches around solid obstacles. This small potion of inaccurate kernels can result in slow convergence or even divergence if the SPD property is locally violated. We suspect that this behavior might come from SDF patches that are not well represented on the original dataset and the limited capability of the simple MLPs architectures employed. Therefore an improved network design and better sampling of the input examples during training may relieve this issue. Furthermore, when testing our model on scenes with unseen shapes (e. g. , triangles) representing interior Neumann boundaries, our model also demonstrates inferior performance (Figure  11 ). We notice, however, that our method can be further fine-tuned for these scenarios by performing additional optimization iterations through Equation (15) (Figure  7  (d)).
Future Work We plan to extend our work to support higher resolutions and 3-D settings. Moreover, our method could be applied to handle Poisson equations on arbitrary meshes, which would require more sophisticated numerical methods and efficient implementation of the downsampling and upsampling operators. Lastly, our method could offer significant speed-ups when dealing with varying boundaries (moving solids or liquids simulations). These scenarios were not explored in this paper; however, we believe that the method can be extended to handle such cases, as long as the training dataset represents moving boundaries accurately. These extensions would greatly increase the application of the proposed approach, since solving the Poisson equation is a widely pervasive problem.
The ATLAS Collaboration The ATLAS experiment at the Large Hadron Collider has a broad physics programme ranging from precision measurements to direct searches for new particles and new interactions, requiring ever larger and ever more accurate datasets of simulated Monte Carlo events. Detector simulation with G 4 is accurate but requires significant CPU resources. Over the past decade, ATLAS has developed and utilized tools that replace the most CPU-intensive component of the simulation -the calorimeter shower simulation -with faster simulation methods. Here, AtlFast3, the next generation of high-accuracy fast simulation in ATLAS is introduced. AtlFast3 combines parameterized approaches with machine-learning techniques and is deployed to meet current and future computing challenges and simulation needs of the ATLAS experiment.
Datasets and samples are presented in Section 3. Sections 4 and 5 discuss the FastCaloSim V2 and FastCaloGAN parameterizations, respectively. The simulation of muon punch-through is discussed in Section 6. Section 7 discusses how AtlFast3 is constructed by combining FastCaloSim V2 and FastCaloGAN. The performance of AtlFast3 in the reconstruction and in physics analysis is discussed in Section 8. Section 9 concludes the paper.
Simulation of reference samples with G 4 The reference samples used to generate the calorimeter parameterizations discussed in this paper were produced using G 4 version 10. 1. 3 [4] , which was released in 2016, and use the FTFP_BERT_ATL physics list  [12]. FTFP_BERT_ATL uses the Bertini intra-nuclear cascade model below 9 GeV and transitions to the Fritiof model  [13] [14] [15]  with a pre-compound model for 12 GeV and higher. The standard configuration for ATLAS simulation in physics analyses, referred to as the MC16 campaign, uses a technique known as Frozen Showers  [5]  to simulate electromagnetic showers in the FCal. This latter configuration has been used for full simulation samples for all papers on Run 2 data published by ATLAS to date. The reference samples do not use the Frozen Shower technique, but it is used for the validation samples discussed in Section 3.
3. The reference datasets consist of single-particle events produced on a cylinder with  = 1148 mm and || < 3550 mm located just outside the TRT because only particles reaching the calorimeters have to be parameterized. This means that the impact of the cryostat and solenoid material is included in the reference datasets. The particles are produced with directions of propagation consistent with production at the interaction point and simulated without the spread of the LHC beam bunches, for positive and negative values of  and with a uniform distribution in. The impact of the spread of the beam bunches is negligible because particles are parameterized according to where they enter the calorimeter. Photons () and electrons ( ± ) are used to parameterize electromagnetic showers, and positively and negatively charged pions ( ± ) are used to parameterize hadronic showers.
Voxelization The spatial energy deposits (hits) in each layer in the G 4 datasets are grouped into volumes called 'voxels' for parameterization of FastCaloSim V2 and FastCaloGAN. Only layers with a significant amount of energy, referred to as 'relevant layers', are considered in the parameterization. Relevant layers are defined using criteria on the fraction of energy deposited in the layers with respect to the total energy deposited in the calorimeters. The criteria used by the two simulators to determine the relevant layers are slightly different: • In FastCaloSim V2, only layers with energy fractions larger than 0. 1% are used; this procedure is performed for each sample independently. • In FastCaloGAN all samples in the same  slice are processed with the same number of voxels. The relevant layers are determined using only the 1 TeV energy point and have an energy fraction larger than 0.
Validation datasets A range of different Monte Carlo samples commonly used for physics performance studies and physics analysis are used to validate the performance of AtlFast3. Table  3  summarizes the key samples, which are also discussed in this section. The matrix element (ME) order describes the precision at which the process is produced by the generator in perturbative quantum chromodynamics (QCD); this can be leading order (LO), next-to-leading order (NLO) or next-to-next-to-leading order (NNLO). The production of  t events was performed with P B r2330  [16]  interfaced with P 6. 427  [17]  for the parton shower and hadronization modelling with the CT10  [18]  set of parton distribution functions (PDFs) and the Perugia2012 set of tuned parameters (P2012 tune)  [19]. At least one of the top quarks is required to produce a lepton when decaying.
, 2015). Sparsity could be considered a weak prior on the distribution of the wavelet coefficients of the solution. This comes from the fact that most images could be represented in a more compressed manner in the wavelet domain. The advancement in deep learning over the recent past has presented encouraging outcomes in the field of deconvolution  (Xu et al. , 2014). Within the astrophysics community, deep learning approaches have been introduced to carry out model fitting that could be considered as a parametric deconvolution  (Tuccillo et al. , 2018). Particularly, Unets  (Ronneberger et al. , 2015)  have gathered tremendous attention due to their effectual performance, which can be attributed to their highly non-linear processing and the availability of large training datasets. Based on Unet,  Sureau et al.
(2020) , but by substituting the Unet denoiser by Learnlet. We compare our results on images extracted from the CANDELS survey  (Grogin et al. , 2011; Koekemoer et al. , 2011). In Section 2, we introduce the deconvolution problem and the deep learningmethods that have been developed to tackle them. In Section 3, we discuss the concept of Learnlet decomposition as introduced by  Ramzi et al. (2021) , and extend the idea to use the network for image deconvolution. We detail out the process of generating our dataset and perfoming the numerical experiments in Section 4, and in Section 5, we demonstrate the results obtained for these experiments. Finally, in Section 6, we conclude our work.
, 2021; Liaudat et al. , 2022 ). • Many galaxies share very similar morphologies. Having a learning process allows to capture these morphologies and improve the deconvolution. However, it also has a few drawbacks: • Generalization properties are not so good. It was shown that Unet does not perform well on image morphologies or noise levels that were not included in the training dataset  (Ramzi et al. , 2021 ). • It is not clear how Unets deal with images containing noise with non-white statistical properties (for e. g. : Poisson, non-stationary Gaussian noise), while sparsity-based techniques can easily consider different kinds of noise  (Starck et al. , 2015). • There is no theory to support the method. In a denoising framework, the Learnlet decomposition has shown interesting properties that could also be considered for deconvolution.
Using a deeper Unet In addition to the compact XDense Unet, we also tested the performance of Unet-64, a deeper Unet as used in  Ramzi et al. (2021)  that has 31,023,940 trainable parameters (around 170 times more parameters than XDense Unet). The architecture is the same as shown in Figure  1. As in the case of Tikhonet, we used the bias-free Unet architecture, as these biases lead to a low generalisation capacity  (Mohan et al. , 2020). The full Python implementation of the code is publicly available 1. 3 Learnlet deconvolution.
θ s (i) are the convolution kernels all of the same square size. • u is an upsampling operation performed by a bicubic interpolator. To make Learnlets behave as closely as possible to wavelets and hence more intuitive,  Ramzi et al. (2021)  constrain the thresholding levels to lie within [0, 5] and force the analysis filters to have a unit norm. Figure  2  shows a schematic of the Learnlet architecture. The full Python implementation of the code is publicly available 2.
Ground truths-CANDELS The Cosmic Assembly Near-IR Deep Extragalactic Legacy Survey, CANDELS  (Grogin et al. , 2011; Koekemoer et al. , 2011) , was conceived to record the first third of galactic evolution from z = 8 to 1. 5 via deep imaging of more than 250,000 galaxies. The entire survey consists of five different image mosaics (GOODS-N, GOODS-S, EGS, UDS, COSMOS), each covering different regions of the sky. To generate our training dataset, using Python, we extracted cutout windows of dimensions 128 × 128 pixels from the CANDELS FITS image mosaics in the F606W filter (V-band) by centering them at the object centroid, without any dynamic range-scaling. For larger objects, we adaptively increased the cutout window dimensions to completely enclose them. In that case, if the window dimensions exceeded 128 × 128 pixels, we downsampled the images to 128 × 128.
Simulations All the extracted ground truth images are convolved with a Gaussian PSF with an FWHM of 15 pixels, which visually blurs the images such that individual small-scale structures are lost. As seen from Eq. 3, since the algorithm takes the PSF as an input parameter and performs Tikhonov deconvolution in the first step, the method would work with any other kind of blurring kernel. The actual performance of the method is mainly dependent on the neural network training, which corresponds to the denoising step. After convolution, we add white Gaussian noise with a standard deviation σ noise having a value such that the faintest object in our dataset has a peak SNR close to 1 and is hence barely visible. For this value of σ noise , we observe a range of SNR values depending on the magnitude of the galaxy.
Pre-processing Each image x (i) was normalised by subtracting its own mean μ (i) and scaling within the [-1, 1] range as follows: x (i) -μ (i) max [x t (i) -μ t (i) ] (8) where x t (i) and μ t (i) denote the i th target image and its corresponding mean. The denominator ensures that all target images have a maximum pixel intensity of 1, and that all noisy images are scaled with respect to their corresponding targets such that the ratios of the peak intensities between them remain the same before and after normalization. Since the flux of the target image is spread out by the PSF as a result of convolution, the noisy images have a lower peak intensity than their corresponding target images. One of the main goals of astrophysical deconvolution is to recover this drop in peak intensity. During training, we augment our dataset by performing random rotations in multiples of 90°, translations and flips along horizontal and vertical axes.
Training After performing a Tikhonov deconvolution as described by Eq. 3, the networks were trained to learn the mapping from the Tikhonov outputs x to the ground truth images x t while minimizing the MSE (Eq. 2). The images were fed to the networks in mini-batches of size 32. All networks were trained until convergence was achieved. We trained the networks with an Adam optimizer  (Kingma and Ba, 2014)  and an initial learning rate of 10 -3 , which was then decreased by half every 25 epochs until it reached a minimum of 10 -5. The way we generate our dataset (as described in Section 4. 1) results in images with varying noise levels, which ensures that the training is not biased by a certain noise level. Table  1  shows a performance comparison of the three methods.