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This paper investigates the use of Mean Opinion Score (MOS), a common image quality metric, as a user-centric evaluation metric for XAI post-hoc explainers. To measure the MOS, a user experiment is proposed, which has been conducted with explanation maps of intentionally distorted images. Three methods from the family of feature attribution methods - Gradient-weighted Class Activation Mapping (Grad-CAM), Multi-Layered Feature Explanation Method (MLFEM), and Feature Explanation Method (FEM) - are compared with this metric. Additionally, the correlation of this new user-centric metric with automatic metrics is studied via Spearman's rank correlation coefficient. MOS of MLFEM shows the highest correlation with automatic metrics of Insertion Area Under Curve (IAUC) and Deletion Area Under Curve (DAUC). However, the overall correlations are limited, which highlights the lack of consensus between automatic and user-centric metrics., Comment: Supported by organization Laboratoire Bordelais de Recherche en Informatique, 15 pages, 4 figures, 3 tables

Aims: Ultraviolet (UV) lines of neutral carbon observed in absorption in the local diffuse interstellar medium (ISM) have long revealed that a substantial fraction of the mass of the gas lies at a thermal pressure one to three orders of magnitude above that of the bulk of the ISM. In this paper, we propose that this enigmatic component originates from shocks propagating at intermediate ($V_S > 30$ km s$^{-1}$) and high velocities ($V_S \geqslant 100$ km s$^{-1}$) in the Warm Neutral Medium (WNM). Methods: Shock waves irradiated by the standard interstellar radiation field (ISRF) are modelled using the Paris-Durham shock code designed to follow the dynamical, thermal, and chemical evolutions of shocks with velocities up to 500 km s$^{-1}$. Each observed line of sight is decomposed into a high pressure and a low pressure components. The column density of carbon at high pressure is confronted to the model predictions to derive the number of shocks along the line of sight and their total dissipation rate. Results: Phase transition shocks spontaneously lead to the presence of high pressure gas in the diffuse ISM and are found to naturally produce neutral carbon with excitation conditions and linewidths in remarkable agreement with the observations. The amounts of neutral carbon at high pressure detected over a sample of 89 lines of sight imply a dissipation rate of mechanical energy with a median of $\sim 3x10^{-25}$ erg cm$^{-3}$ s$^{-1}$ and a dispersion of about a factor of three. This distribution of the dissipation rate weakly depends on the detailed characteristics of shocks as long as they propagate at velocities between 30 and 200 kms s$^{-1}$ in a medium with a preshock density $n_H^0 \ge 0.3$ cm s$^{-3}$ and a transverse magnetic field $B_0 \leqslant 3$ $\mu$G. We not only show that this solution is consistent with a scenario of shocks driven by supernovae remnants (SNR) but also that this scenario is, in fact, unavoidable. Any line of sight in the observational sample is bound to intercept SNRs, mostly distributed in the spiral arms of the Milky Way, and expanding in the diffuse ionized and neutral phases of the Galaxy. Surprisingly, the range of dissipation rate derived here, in events that probably drive turbulence in the WNM, is found to be comparable to the distribution of the kinetic energy transfer rate of the turbulent cascade derived from the observations of CO in the Cold Neutral Medium (CNM). Conclusions: This work reveals a possible direct tracer of the mechanisms by which mechanical energy is injected in the ISM. It also suggests that a still unknown connection exists between the amount of energy dissipated during the injection process in the WNM and that used to feed interstellar turbulence and the turbulent cascade observed in the CNM., Comment: Astronomy and Astrophysics - A&A, In press

Context: Atomic and molecular lines in galaxies offer insights into energy budgets and feedback mechanisms. Aims: This study establishes a new framework for interpreting these lines and deducing energy budgets from observations. Methods: Atomic and molecular lines detected in a given object are assumed to result from the combination of distributions of shocks and photo-dissociation regions (PDR). Using the Paris-Durham shock code and the Meudon PDR code, emissions are computed over a wide range of parameters. Total emissions are calculated using probability distribution functions, with a defined distance metric based on observed and predicted intensity ratios. Results: We analyze the radio galaxy 3C 326 N, finding both shocks and PDRs necessary to explain the line fluxes. Viable solutions occur only at low densities ($\rm n_H < 100 cm^{-3}$), indicating emission from diffuse interstellar matter. The optimal solution involves low-velocity shocks (5-20 km/s) in PDRs illuminated by UV radiation ten times stronger than in the solar neighborhood. The H$2$ 0-0 S(0) $28 \mu$m, [CII] $158 \mu$m, and [OI] $63 \mu$m lines originate from PDRs, while other H$2$ lines mostly come from shocks. The reprocessed radiative and mechanical energies are $\rm {L_{UV} = 6.3\times10^9 L\odot}$ and $\rm {L_K = 3.9\times10^8 L_\odot}$, respectively, in agreement with 3C 326 N's infrared luminosity, and consistent with 1% of the AGN jet kinetic power dissipated in the interstellar medium. Conclusions: This study demonstrates that the radiative and mechanical energy budgets of galaxies can be derived from observations of atomic and molecular lines alone. It highlights the unexpected importance of the diffuse medium for 3C 326 N. Comparison with new JWST data for 3C 326 N shows striking agreement, opening new prospects for predicting and interpreting extragalactic observations., Comment: Accepted for publication in A&A, 22 pages, 15 figures

Context. Atomic and molecular line emissions from shocks may provide valuable information on the injection of mechanical energy in the interstellar medium (ISM), the generation of turbulence, and the processes of phase transition between the Warm Neutral Medium (WNM) and the Cold Neutral Medium (CNM).Aims. In this series of papers, we investigate the properties of shocks propagating in the WNM. Our objective is to identify the tracers of these shocks, use them to interpret ancillary observations of the local diffuse matter, and provide predictions for future observations.Methods. Shocks propagating in the WNM are studied using the Paris-Durham shock code, a multi-fluid model built to follow the thermodynamical and chemical structures of shock waves, at steady-state, in a plane-parallel geometry. The code, already designed to take into account the impact of an external radiation field, is updated to treat self-irradiated shocks at intermediate (30, Comment: Astronomy and Astrophysics - A&A (2024)

Foundation models are powerful technologies: how they are released publicly directly shapes their societal impact. In this position paper, we focus on open foundation models, defined here as those with broadly available model weights (e.g. Llama 2, Stable Diffusion XL). We identify five distinctive properties (e.g. greater customizability, poor monitoring) of open foundation models that lead to both their benefits and risks. Open foundation models present significant benefits, with some caveats, that span innovation, competition, the distribution of decision-making power, and transparency. To understand their risks of misuse, we design a risk assessment framework for analyzing their marginal risk. Across several misuse vectors (e.g. cyberattacks, bioweapons), we find that current research is insufficient to effectively characterize the marginal risk of open foundation models relative to pre-existing technologies. The framework helps explain why the marginal risk is low in some cases, clarifies disagreements about misuse risks by revealing that past work has focused on different subsets of the framework with different assumptions, and articulates a way forward for more constructive debate. Overall, our work helps support a more grounded assessment of the societal impact of open foundation models by outlining what research is needed to empirically validate their theoretical benefits and risks.

We present a novel method for establishing large data local well-posedness in low regularity Sobolev spaces for general quasilinear Schr\"odinger equations with non-degenerate and nontrapping metrics. Our result represents a definitive improvement over the landmark results of Kenig, Ponce, Rolvung and Vega, as it weakens the regularity and decay assumptions to the same scale of spaces considered by Marzuola, Metcalfe, and Tataru, but removes the uniform ellipticity assumption on the metric from their result. Our method has the additional benefit of being relatively simple but also very robust. In particular, it only relies on the use of pseudodifferential calculus for classical symbols., Comment: 52 pages

In the field of Explainable AI, multiples evaluation metrics have been proposed in order to assess the quality of explanation methods w.r.t. a set of desired properties. In this work, we study the articulation between the stability, correctness and plausibility of explanations based on feature importance for image classifiers. We show that the existing metrics for evaluating these properties do not always agree, raising the issue of what constitutes a good evaluation metric for explanations. Finally, in the particular case of stability and correctness, we show the possible limitations of some evaluation metrics and propose new ones that take into account the local behaviour of the model under test.

Gaia's readout window strategy is challenged by very dense fields in the sky. Therefore, in addition to standard Gaia observations, full Sky Mapper (SM) images were recorded for nine selected regions in the sky. A new software pipeline exploits these Service Interface Function (SIF) images of crowded fields (CFs), making use of the availability of the full two-dimensional (2D) information. This new pipeline produced half a million additional Gaia sources in the region of the omega Centauri ($\omega$ Cen) cluster, which are published with this Focused Product Release. We discuss the dedicated SIF CF data reduction pipeline, validate its data products, and introduce their Gaia archive table. Our aim is to improve the completeness of the {\it Gaia} source inventory in a very dense region in the sky, $\omega$ Cen. An adapted version of {\it Gaia}'s Source Detection and Image Parameter Determination software located sources in the 2D SIF CF images. We validated the results by comparing them to the public {\it Gaia} DR3 catalogue and external Hubble Space Telescope data. With this Focused Product Release, 526\,587 new sources have been added to the {\it Gaia} catalogue in $\omega$ Cen. Apart from positions and brightnesses, the additional catalogue contains parallaxes and proper motions, but no meaningful colour information. While SIF CF source parameters generally have a lower precision than nominal {\it Gaia} sources, in the cluster centre they increase the depth of the combined catalogue by three magnitudes and improve the source density by a factor of ten. This first SIF CF data publication already adds great value to the {\it Gaia} catalogue. It demonstrates what to expect for the fourth {\it Gaia} catalogue, which will contain additional sources for all nine SIF CF regions.

Context. Strongly lensed quasars are fundamental sources for cosmology. The Gaia space mission covers the entire sky with the unprecedented resolution of $0.18$" in the optical, making it an ideal instrument to search for gravitational lenses down to the limiting magnitude of 21. Nevertheless, the previous Gaia Data Releases are known to be incomplete for small angular separations such as those expected for most lenses. Aims. We present the Data Processing and Analysis Consortium GravLens pipeline, which was built to analyse all Gaia detections around quasars and to cluster them into sources, thus producing a catalogue of secondary sources around each quasar. We analysed the resulting catalogue to produce scores that indicate source configurations that are compatible with strongly lensed quasars. Methods. GravLens uses the DBSCAN unsupervised clustering algorithm to detect sources around quasars. The resulting catalogue of multiplets is then analysed with several methods to identify potential gravitational lenses. We developed and applied an outlier scoring method, a comparison between the average BP and RP spectra of the components, and we also used an extremely randomised tree algorithm. These methods produce scores to identify the most probable configurations and to establish a list of lens candidates. Results. We analysed the environment of 3 760 032 quasars. A total of 4 760 920 sources, including the quasars, were found within 6" of the quasar positions. This list is given in the Gaia archive. In 87\% of cases, the quasar remains a single source, and in 501 385 cases neighbouring sources were detected. We propose a list of 381 lensed candidates, of which we identified 49 as the most promising. Beyond these candidates, the associate tables in this Focused Product Release allow the entire community to explore the unique Gaia data for strong lensing studies further., Comment: 35 pages, 60 figures, accepted for publication by Astronomy and Astrophysics

The third Gaia Data Release (DR3) provided photometric time series of more than 2 million long-period variable (LPV) candidates. Anticipating the publication of full radial-velocity (RV) in DR4, this Focused Product Release (FPR) provides RV time series for a selection of LPVs with high-quality observations. We describe the production and content of the Gaia catalog of LPV RV time series, and the methods used to compute variability parameters published in the Gaia FPR. Starting from the DR3 LPVs catalog, we applied filters to construct a sample of sources with high-quality RV measurements. We modeled their RV and photometric time series to derive their periods and amplitudes, and further refined the sample by requiring compatibility between the RV period and at least one of the $G$, $G_{\rm BP}$, or $G_{\rm RP}$ photometric periods. The catalog includes RV time series and variability parameters for 9\,614 sources in the magnitude range $6\lesssim G/{\rm mag}\lesssim 14$, including a flagged top-quality subsample of 6\,093 stars whose RV periods are fully compatible with the values derived from the $G$, $G_{\rm BP}$, and $G_{\rm RP}$ photometric time series. The RV time series contain a mean of 24 measurements per source taken unevenly over a duration of about three years. We identify the great most sources (88%) as genuine LPVs, with about half of them showing a pulsation period and the other half displaying a long secondary period. The remaining 12% consists of candidate ellipsoidal binaries. Quality checks against RVs available in the literature show excellent agreement. We provide illustrative examples and cautionary remarks. The publication of RV time series for almost 10\,000 LPVs constitutes, by far, the largest such database available to date in the literature. The availability of simultaneous photometric measurements gives a unique added value to the Gaia catalog (abridged), Comment: 36 pages, 38 figures

We provide a complete local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension on a connected fluid domain. Our well-posedness theory includes: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: Our uniqueness result holds at the level of the Lipschitz norm of the velocity and the $C^{1,\frac{1}{2}}$ regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove refined, essentially scale invariant energy estimates for solutions, relying on a newly constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a sharp continuation criterion in the physically relevant pointwise norms, at the level of scaling. In essence, we show that solutions can be continued as long as the velocity is in $L_T^1W^{1,\infty}$ and the free surface is in $L_T^1C^{1,\frac{1}{2}}$, which is at the same level as the Beale-Kato-Majda criterion for the boundaryless case; (vi) A novel proof of the construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in more general fluid domains., Comment: 117 pages, some typos corrected

(Abridged) H2 is the most abundant molecule in the Universe. Thanks to its widely spaced energy levels, it predominantly lights up in warm gas, T > 100 K, such as shocked regions, and it is one of the key targets of JWST observations. These include shocks from protostellar outflows, all the way up to starburst galaxies and AGN. Shock models are able to simulate H2 emission. We aim to explore H2 excitation using such models, and to test over which parameter space distinct signatures are produced in H2 emission. We present simulated H2 emission using the Paris-Durham shock code over an extensive grid of 14,000 plane-parallel stationary shock models, a large subset of which are exposed to an external UV radiation field. The grid samples 6 input parameters: preshock density, shock velocity, transverse magnetic field strength, UV radiation field strength, cosmic-ray-ionization rate, and PAH abundance. Physical quantities, such as temperature, density, and width, have been extracted along with H2 integrated line intensities. The strength of the transverse magnetic field, set by the scaling factor, b, plays a key role in the excitation of H2. At low values of b (<~ 0.3, J-type shocks), H2 excitation is dominated by vibrationally excited lines; at higher values (b >~ 1, C-type shocks), rotational lines dominate the spectrum for shocks with an external radiation field comparable to (or lower than) the solar neighborhood. Shocks with b >= 1 can be spatially resolved with JWST for nearby objects. When the input kinetic energy flux increases, the excitation and integrated intensity of H2 increases similarly. An external UV field mainly serves to increase the excitation, particularly for shocks where the input radiation energy is comparable to the input kinetic energy flux. These results provide an overview of the energetic reprocessing of input kinetic energy flux and the resulting H2 line emission., Comment: Published in A&A

Energy system optimization models are important tools to provide insights regarding trade-offs and interrelations in cost-efficient transformation pathways towards a climate neutral energy system. Using an optimization model of the European electricity system, we study the influence of either an emission cap or temporally resolved minimum renewable generation shares as means to power system decarbonization. Such minimum shares serve as a stylized representation of novel, more granular procurement or certification schemes for renewable power generation. We observe that decarbonization through minimum shares results in a significant increase in system costs compared to the least cost solution obtained through a direct implementation of the corresponding emission reduction as a cap. Nevertheless, these shares provide direct incentives for expanding renewable generation and storage technologies, thus acting as an additional, potentially more robust decarbonization mechanism.

A central role in shaping the experience of users online is played by recommendation algorithms. On the one hand they help retrieving content that best suits users taste, but on the other hand they may give rise to the so called "filter bubble" effect, favoring the rise of polarization. In the present paper we study how a user-user collaborative-filtering algorithm affects the behavior of a group of agents repeatedly exposed to it. By means of analytical and numerical techniques we show how the system stationary state depends on the strength of the similarity and popularity biases, quantifying respectively the weight given to the most similar users and to the best rated items. In particular, we derive a phase diagram of the model, where we observe three distinct phases: disorder, consensus and polarization. In the latter users spontaneously split into different groups, each focused on a single item. We identify, at the boundary between the disorder and polarization phases, a region where recommendations are nontrivially personalized without leading to filter bubbles. Finally, we show that our model can reproduce the behavior of users in the online music platform last.fm. This analysis paves the way to a systematic analysis of recommendation algorithms by means of statistical physics methods and opens to the possibility of devising less polarizing recommendation algorithms.

XMM-Newton, a European Space Agency observatory, has been observing the X-ray, ultra-violet and optical sky for 23 years. During this time, astronomy has evolved from mainly studying single sources to populations and from a single wavelength, to multi-wavelength or messenger data. We are also moving into an era of time domain astronomy. New software and methods are required to accompany evolving astronomy and prepare for the next generation X-ray observatory, Athena. Here we present XMM2ATHENA, a programme funded by the European Union's Horizon 2020 research and innovation programme. XMM2ATHENA builds on foundations laid by the XMM-Newton Survey Science Centre (XMM-SSC), including key members of this consortium and the Athena Science ground segment, along with members of the X-ray community. The project is developing and testing new methods and software to allow the community to follow the X-ray transient sky in quasi-real time, identify multi-wavelength or messenger counterparts of XMM-Newton sources and determine their nature using machine learning. We detail here the first milestone delivery of the project, a new online, sensitivity estimator. We also outline other products, including the forthcoming innovative stacking procedure and detection algorithms to detect the faintest sources. These tools will then be adapted for Athena and the newly detected or identified sources will enhance preparation for observing the Athena X-ray sky., Comment: 7 pages, 1 figure, accepted for publication in Astronomische Nachrichten

Sports video analysis is a widespread research topic. Its applications are very diverse, like events detection during a match, video summary, or fine-grained movement analysis of athletes. As part of the MediaEval 2022 benchmarking initiative, this task aims at detecting and classifying subtle movements from sport videos. We focus on recordings of table tennis matches. Conducted since 2019, this task provides a classification challenge from untrimmed videos recorded under natural conditions with known temporal boundaries for each stroke. Since 2021, the task also provides a stroke detection challenge from unannotated, untrimmed videos. This year, the training, validation, and test sets are enhanced to ensure that all strokes are represented in each dataset. The dataset is now similar to the one used in [1, 2]. This research is intended to build tools for coaches and athletes who want to further evaluate their sport performances., Comment: MediaEval 2022 Workshop, Jan 2023, Bergen, Norway. arXiv admin note: substantial text overlap with arXiv:2112.11384

The most popular methods in AI-machine learning paradigm are mainly black boxes. This is why explanation of AI decisions is of emergency. Although dedicated explanation tools have been massively developed, the evaluation of their quality remains an open research question. In this paper, we generalize the methodologies of evaluation of post-hoc explainers of CNNs' decisions in visual classification tasks with reference and no-reference based metrics. We apply them on our previously developed explainers (FEM, MLFEM), and popular Grad-CAM. The reference-based metrics are Pearson correlation coefficient and Similarity computed between the explanation map and its ground truth represented by a Gaze Fixation Density Map obtained with a psycho-visual experiment. As a no-reference metric, we use stability metric, proposed by Alvarez-Melis and Jaakkola. We study its behaviour, consensus with reference-based metrics and show that in case of several kinds of degradation on input images, this metric is in agreement with reference-based ones. Therefore, it can be used for evaluation of the quality of explainers when the ground truth is not available., Comment: Due to a bug found in the code, all tables and figures were redone. The new results did not change the main conclusion, except for the best explainer. FEM has performed better than MLFEM; 25 pages, 16 tables, 16 figures; Submitted to "Advances in Artificial Intelligence and Machine Learning" (ISSN: 2582-9793)

Transformer language models encode the notion of word order using positional information. Most commonly, this positional information is represented by absolute position embeddings (APEs), that are learned from the pretraining data. However, in natural language, it is not absolute position that matters, but relative position, and the extent to which APEs can capture this type of information has not been investigated. In this work, we observe that models trained with APE over-rely on positional information to the point that they break-down when subjected to sentences with shifted position information. Specifically, when models are subjected to sentences starting from a non-zero position (excluding the effect of priming), they exhibit noticeably degraded performance on zero to full-shot tasks, across a range of model families and model sizes. Our findings raise questions about the efficacy of APEs to model the relativity of position information, and invite further introspection on the sentence and word order processing strategies employed by these models., Comment: Accepted at EMNLP 2022 Findings; 5 pages and 15 pages Appendix

In this paper, we describe a universal method for extracting the underlying monotonic trend factor from time series data. We propose an approach related to the Mann-Kendall test, a standard monotonic trend detection method and call it contrastive trend estimation (CTE). We show that the CTE method identifies any hidden trend underlying temporal data while avoiding the standard assumptions used for monotonic trend identification. In particular, CTE can take any type of temporal data (vector, images, graphs, time series, etc.) as input. We finally illustrate the interest of our CTE method through several experiments on different types of data and problems.

Let $(\Omega,\Sigma,\mu)$ be a measure space, and $1\leq p\leq \infty$. A subspace $E\subseteq L_p(\mu)$ is said to do stable phase retrieval (SPR) if there exists a constant $C\geq 1$ such that for any $f,g\in E$ we have $$ \inf_{|\lambda|=1} \|f-\lambda g\|\leq C\||f|-|g|\|. $$ In this case, if $|f|$ is known, then $f$ is uniquely determined up to an unavoidable global phase factor $\lambda$; moreover, the phase recovery map is $C$-Lipschitz. Phase retrieval appears in several applied circumstances, ranging from crystallography to quantum mechanics. In this article, we construct various subspaces doing stable phase retrieval, and make connections with $\Lambda(p)$-set theory. Moreover, we set the foundations for an analysis of stable phase retrieval in general function spaces. This, in particular, allows us to show that H\"older stable phase retrieval implies stable phase retrieval, improving the stability bounds in a recent article of M. Christ and the third and fourth authors. We also characterize those compact Hausdorff spaces $K$ such that $C(K)$ contains an infinite dimensional SPR subspace., Comment: 58 pages

Aims: The large abundances of CH$^+$ in the diffuse interstellar medium (ISM) are a long standing issue of our understanding of the thermodynamical and chemical states of the gas. We investigate, here, the formation of CH+ in turbulent and multiphase environments, where the heating of the gas is almost solely driven by the photoelectric effect. Methods: The diffuse ISM is simulated using the magnetohydrodynamic (MHD) code RAMSES which self-consistently computes the dynamical and thermal evolution of the gas along with the time-dependent evolutions of the abundances of H$^+$, H, and H$_2$. The rest of the chemistry, including the abundance of CH$^+$, is computed in post-processing, at equilibrium, under the constraint of out-ofequilibrium of H$^+$, H, and H$_2$. The comparison with the observations is performed taking into account an often neglected, yet paramount, piece of information, namely the length of the intercepted diffuse matter along the observed lines of sight. Results: The quasi totality of the mass of CH$^+$ originates from the unstable gas, in environments where the kinetic temperature is larger than 600 K, the density ranges between 0.6 and 10 cm$^{-3}$, the electronic fraction ranges between 3 x 10$^{-4}$ and 6 x 10$^{-3}$, and the molecular fraction is smaller than 0.4. Its formation is driven by warm and out-of-equilibrium H$_2$ initially formed in the cold neutral medium (CNM) and injected in more diffuse environments and even the warm neutral medium (WNM) through a combination of advection and thermal instability. The simulation which displays the tightest agreement with the HI-to-H$_2$ transition and the thermal pressure distribution observed in the Solar Neighborhood is found to naturally reproduce the observed abundances of CH$^+$, the dispersion of observations, the probability of occurrence of most of the lines of sight, the fraction of non-detections of CH$^+$, and the distribution of its line profiles. The amount of CH$^+$ and the statistical properties of the simulated lines of sight are set by the fraction of unstable gas rich in H$_2$ which is controlled, on Galactic scales, by the mean density of the diffuse ISM (or, equivalently, its total mass), the amplitude of the mean UV radiation field, and the strength of the turbulent forcing. Conclusions: This work offers a new and natural solution to an 80 years old chemical riddle. The almost ubiquitous presence of CH$^+$ in the diffuse ISM likely results from the exchanges of matter between the CNM and the WNM induced by the combination of turbulent advection and thermal instability, without the need to invoke ambipolar diffusion or regions of intermittent turbulent dissipation. Through two phase turbulent mixing, CH$^+$ might thus be a tracer of the H$_2$ mass loss rate of CNM clouds., Comment: Astronomy and Astrophysics, EDP Sciences, In press