Nonlinear Scaling Research Articles

Rotational Excitation Cross Sections for Chloronium Based on a New 5D Interaction Potential with Molecular Hydrogen.

Chloronium (H2Cl+) is an important intermediate of Cl-chemistry in space. The accurate knowledge of its collisional properties allows a better interpretation of the corresponding observations in interstellar clouds and, therefore, a better estimation of its abundance in these environments. While the ro-vibrational spectroscopy of H2Cl+ is well-known, the studies of its collisional excitation are rather limited and these are available for the interaction with helium atoms only. We provide a new 5-dimensional rigid rotor potential energy surface for the interaction of H2Cl+ with H2, calculated from explicitly correlated coupled cluster ab initio theory, which was fitted then with a set of analytical functions, allowing to perform scattering calculations using accurate quantum theories. We analyze the collision-energy dependence of the rotational state-to-state cross sections and the temperature-dependence of the corresponding thermal rate coefficients, with particular attention on the collisional propensity rules. When comparing our results for collisions with H2 with those obtained with He as a colliding partner, we found very significant differences with nonlinear scaling trends, which proves again that He is not a suitable proxy for collisions between hydride molecules and molecular hydrogen, the most abundant gas particle in the interstellar medium.

A field-level emulator for modified gravity

Abstract Stage IV surveys like LSST and Euclid present a unique opportunity to shed light on the nature of dark energy. However, their full constraining power cannot be unlocked unless accurate predictions are available at all observable scales. Currently, only the linear regime is well understood in models beyond ΛCDM: on the nonlinear scales, expensive numerical simulations become necessary, whose direct use is impractical in the analyses of large datasets. Recently, machine learning techniques have shown the potential to break this impasse: by training emulators, we can predict complex data fields in a fraction of the time it takes to produce them. In this work, we present a field-level emulator capable of turning a ΛCDM N-body simulation into one evolved under f(R) gravity. To achieve this, we build on the map2map neural network, using the strength of modified gravity $|f_{R_0}|$ as style parameter. We find that our emulator correctly estimates the changes it needs to apply to the positions and velocities of the input N-body particles to produce the target simulation. We test the performance of our network against several summary statistics, finding $1{{\%}}$ agreement in the power spectrum up to k ∼ 1 h Mpc−1, and $1.5{{\%}}$ agreement against the independent boost emulator eMantis. Although the algorithm is trained on fixed cosmological parameters, we find it can extrapolate to models it was not trained on. Coupled with available field-level emulators and simulation suites for ΛCDM, our algorithm can be used to constrain modified gravity in the large-scale structure using full information available at the field level.

Psi-GAN: a power-spectrum-informed generative adversarial network for the emulation of large-scale structure maps across cosmologies and redshifts

ABSTRACT Simulations of the dark matter distribution throughout the Universe are essential in order to analyse data from cosmological surveys. N-body simulations are computationally expensive, and many cheaper alternatives (such as lognormal random fields) fail to reproduce accurate statistics of the smaller, non-linear scales. In this work, we present Psi-GAN (power-spectrum-informed generative adversarial network), a machine learning model that takes a two-dimensional lognormal dark matter density field and transforms it into a more realistic field. We construct Psi-GAN so that it is continuously conditional, and can therefore generate realistic realizations of the dark matter density field across a range of cosmologies and redshifts in $z \in [0, 3]$. We train Psi-GAN as a generative adversarial network on $2\, 000$ simulation boxes from the Quijote simulation suite. We use a novel critic architecture that utilizes the power spectrum as the basis for discrimination between real and generated samples. Psi-GAN shows agreement with N-body simulations over a range of redshifts and cosmologies, consistently outperforming the lognormal approximation on all tests of non-linear structure, such as being able to reproduce both the power spectrum up to wavenumbers of $1~h~\mathrm{Mpc}^{-1}$, and the bispectra of target N-body simulations to within ${\sim }5$ per cent. Our improved ability to model non-linear structure should allow more robust constraints on cosmological parameters when used in techniques such as simulation-based inference.

Euclid preparation. Simulations and non-linearities beyond Lambda cold dark matter. I. Numerical methods and validation

To constrain cosmological models beyond Lambda CDM, the development of the analysis pipeline requires simulations that capture the non-linear phenomenology of such models. We present an overview of numerical methods and $N$-body simulation codes developed to study the non-linear regime of structure formation in alternative dark energy and modified gravity theories. We review a variety of numerical techniques and approximations employed in cosmological $N$-body simulations to model the complex phenomenology of scenarios beyond Lambda CDM. This includes discussions on solving non-linear field equations, accounting for fifth forces, and implementing screening mechanisms. Furthermore, we conduct a code comparison exercise to assess the reliability and convergence of different simulation codes across a range of models. Our analysis demonstrates a high degree of agreement among the outputs of different simulation codes, typically within 2<!PCT!> for the predicted modification of the matter power spectrum and within 4<!PCT!> for the predicted modification of the halo mass function, although some approximations degrade accuracy a bit further. This provides confidence in current numerical methods of modelling cosmic structure formation beyond Lambda CDM. We highlight recent advances made in simulating the non-linear scales of structure formation, which are essential for leveraging the full scientific potential of the forthcoming observational data from the mission.

Prioritisation of infectious diseases from a public health perspective: a multi-criteria decision analysis study, France, 2024.

BackgroundWithin the International Health Regulations framework, the French High Council for Public Health was mandated in 2022 by health authorities to establish a list of priority infectious diseases for public health, surveillance and research in mainland and overseas France.AimOur objective was to establish this list.MethodsA multi-criteria decision analysis was used, as recommended by the European Centre for Disease Prevention and Control. A list of 95 entities (infectious diseases or groups of these, including the World Health Organization (WHO)-labelled 'Disease X') was established by 17 infectious disease experts. Ten criteria were defined to score entities: incidence rate, case fatality rate, potential for emergence and spread, impact on the individual, on society, on socially vulnerable groups, on the healthcare system, and need for new preventive tools, new curative therapies, and surveillance. Each criterion was assigned a relative weight by 77 multidisciplinary experts. For each entity, 98 physicians from various specialties rated each criterion against the entity, using a four-class Likert-type scale; the ratings were converted into numeric values with a nonlinear scale and respectively weighted to calculate the entity score.ResultsFifteen entities were ranked as high-priorities, including Disease X and 14 known pathologies (e.g. haemorrhagic fevers, various respiratory viral infections, arboviral infections, multidrug-resistant bacterial infections, invasive meningococcal and pneumococcal diseases, prion diseases, rabies, and tuberculosis).ConclusionThe priority entities agreed with those of the WHO in 2023; almost all were currently covered by the French surveillance and alert system. Repeating this analysis periodically would keep the list updated.

Map-level baryonification: Efficient modelling of higher-order correlations in the weak lensing and thermal Sunyaev-Zeldovich fields

Semi-analytic methods can generate baryon-corrected fields from N-body simulations (“baryonification”) and are rapidly becoming a ubiquitous tool in modeling structure formation on non-linear scales. We extend this formalism to consistently model the weak lensing and thermal Sunyaev-Zeldovich (tSZ) fields directly on the full-sky, with an emphasis on higher-order correlations. We use the auto- and cross- Nth-order moments, with N∈{2,3,4}, as a summary statistic of the lensing and tSZ fields, and show that our model can jointly fit these statistics measured in IllustrisTNG to within measurement uncertainties, for scales above and across multiple redshifts. The model predictions change only minimally when including additional information from secondary halo properties, such as halo concentration and ellipticity. Each individual moment is dependent on halos of different mass ranges and has different sensitivities to the model parameters. A simulation-based forecast on the ULAGAM simulation suite shows that the combination of all moments, measured from current and upcoming lensing and tSZ surveys, can jointly constrain cosmology and baryons to high precision. The lensing and tSZ field are sensitive to different combinations of the baryonification parameters, with degeneracy directions that are often orthogonal, and the combination of the two fields leads to significantly better constraints on both cosmology and astrophysics. Our pipeline for map-level baryonification is publicly available at https://github.com/DhayaaAnbajagane/Baryonification.

A robust feature-based full-field initial value estimation in path-independent digital image correlation for large deformation measurement

The feature-based path-independent digital image correlation (DIC) method has been shown to be a formidable tool for non-contact, full-field measurement of large deformation, but its effectiveness hinges crucially on acquiring sufficient matched features to perform a reliable full-field initial value estimation (IVE) for all points of interest (POIs), thus ensuring their successful and rapid convergence in the succeeding path-independent iterative DIC refinement. This prerequisite is a challenging task particularly when confronted with large deformation. Moreover, in many real-world measurement scenarios, the accuracy of IVE is also influenced by image noise, such as Gaussian noise and shot noise, further compounding the challenge. To mitigate these issues, we propose a robust feature-based full-field IVE method. The core of this method consists of two main components: (i) For feature detection, we leverage the strengths of nonlinear multiscale representations on speckle images using an Accelerated-KAZE (A-KAZE) detector, which extracts features in a nonlinear scale space via nonlinear diffusion filtering. Noise is suppressed and edges are preserved. Compared to existing state-of-the-art feature detectors used in DIC, such as Scale Invariant Feature Transform (SIFT) and Speeded-Up Robust Feature (SURF) detectors, which rely on the use of Gaussian linear scale spaces, the A-KAZE-based nonlinear scale space detector identifies more salient features with higher localization accuracy. (ii) For feature description, considering the need for robustness against large deformation and the computational burden of descriptor matching for considerable salient features that may be detected in speckle images, we introduce a robust Gradient Location and Orientation Histogram (GLOH) descriptor and propose an improved version of it. The GLOH's improved version incorporates a restricted adaptive binning (RAB) strategy to optimize the descriptor’s structure parameters, which is able to reduce the computational cost of descriptor matching through restricting its dimensionality while without sacrificing its robustness and discriminability. These two components are designed to provide sufficient matched features for a full-field IVE. The initial deformation for each POI is estimated independently by fitting a local affine transformation model, which is refined to subpixel accuracy through iterative path-independent DIC analysis. To handle complex large deformation, the inverse compositional Gauss-Newton (IC-GN) algorithm with a second-order shape function is employed. Extensive experimental results demonstrate that our method has improved IVE accuracy as well as behaves more robustness against local geometric transformations and image noise including Gaussian noise and shot noise, as compared to existing state-of-the-art feature-based full-field IVE methods.

Process Simulations and Technoeconomic Analysis of Ethylene Production By Electrochemical Reduction of Carbon Capture Solutions

Electrocatalytic reduction offers an alternate route for production of olefins from non-traditional feedstocks such as CO2, resulting in a lower carbon footprint if coupled with renewable sources of energy.Electrolyzer architectures employing bipolar membranes (BPMs) to convert aqueous alkaline carbonates (carbon capture solution) into hydrocarbons are a potential solution to overcome the limitations of high H2 production and crossover rates that limit CO2 conversion in conventional CO2 gas-fed electrochemical systems. Current literature lacks a full process analysis of such a system for C2+ hydrocarbons production, especially the inclusion of direct air capture of CO2 as part of the process and identification of the key tradeoffs involved in such systems. To address this important issue, here we present comprehensive process simulations, process design analysis, and technoeconomic evaluation of integrated electrolysis-based systems (beginning with CO2 capture and ending in electrolyzer product separation and stream recycling) for ethylene production from carbonates. Process simulations revealed the importance of upstream concentration of the captured carbonate feed stream by a membrane system operable at alkaline conditions, which makes the overall process economically feasible. Electrolyzer sizing, scale-up, and configuration are examined in more detail. Costing of the electrolyzer was approached with nonlinear scaling to better account for economies of scale relevant for electrochemical systems. Three different scenarios were considered for a 2 million metric tons (MMT)/yr ethylene plant. A set of key projected performance metrics has been determined for future profitable large-scale production of ethylene from alkaline carbonates. This BPM-based conversion process was evaluated with both direct air capture (DAC) and flue gas capture scenarios for carbonate production, demonstrating roughly equivalent economic feasibility (<8% difference between the 2 cases). Ongoing improvements in BPM-based processes can lead to ethylene minimum selling prices (MSPs) comparable to MSPs from naphtha-based processes, which are also projected to be lower than those of more conventional CO2 gas-fed electrochemical processes.

Weakly nonlinear analysis of minimal models for Turing patterns

The simplest particle-based mass-action models for Turing instability – i.e. those with only two component species undergoing instantaneous interactions of at most two particles, with the smallest number of distinct interactions – fall into a surprisingly small number of classes of reaction schemes. In previous work we have computed this classification, with different schemes distinguished by the structure of the interactions. Within a given class the reaction stoichiometry and rates remain as parameters that determine the linear and nonlinear evolution of the system.Adopting the usual weakly nonlinear scalings and analysis reveals that, under suitable choices of reaction stoichiometry, and in nine of the 11 classes of minimal scheme exhibiting a spatially in-phase (“true activator-inhibitor”) Turing instability, stable patterns are indeed generated in open regions of parameter space via a generically supercritical bifurcation from the spatially uniform state. In three of these classes the instability is always supercritical while in six there is an open region in which it is subcritical. Intriguingly, however, in the remaining two classes of minimal scheme we require different weakly nonlinear scalings, since the coefficient in the usual cubic normal form unexpectedly vanishes identically. In these cases, a different set of asymptotic scalings is required.We present a complete analysis through deriving the normal form for these two cases also, which involves quintic terms. This fifth-order normal form also captures the behaviour along the boundaries between the supercritical and subcritical cases of the cubic normal form. The details of these calculations reveal the distinct roles played by reaction rate parameters as compared to stoichiometric parameters.We quantitatively validate our analysis via numerical simulations and confirm the two different scalings for the amplitude of predicted stable patterned states.

Understanding the impact of non-linearity in the SIS model

The SIS model is a classic model from epidemiology that formalizes a variety of diffusion processes on networks, such as biological infections and information dissemination. In this model, vertices are either infected or susceptible to an infection. Infected vertices infect their neighbors independently at a rate λ>0, and each infected vertex becomes susceptible at a rate of 1. Overall, these dynamics imply that each susceptible vertex with exactly m infected neighbors becomes infected at rate λm, that is, linear in m. However, it has been observed that various processes exhibit a non-linear scaling of the infection rate with respect to m. For these kinds of processes, no fully rigorous guarantees exist so far.We address this shortcoming by considering a variant of the SIS model in which vertices get infected at a rate that scales polynomially in the number of their infected neighbors, weighted by the infection coefficient λ. We give the first fully rigorous results for thresholds of λ at which the expected survival time becomes super-polynomial. For cliques we show that when the infection rate scales sub-linearly, the threshold only shifts by a poly-logarithmic factor, compared to the standard SIS model. In contrast, super-linear scaling changes the process considerably and shifts the threshold by a polynomial term. For stars, sub-linear and super-linear scaling behave similar and both shift the threshold by a polynomial factor. Our bounds are almost tight, as they are only apart by at most a poly-logarithmic factor from the lower thresholds, at which the expected survival time is logarithmic.

Sensitivity analysis of simulation-based inference for galaxy clustering

ABSTRACT Simulation-based inference (SBI) is a promising approach to leverage high-fidelity cosmological simulations and extract information from the non-Gaussian, non-linear scales that cannot be modelled analytically. However, scaling SBI to the next generation of cosmological surveys faces the computational challenge of requiring a large number of accurate simulations over a wide range of cosmologies, while simultaneously encompassing large cosmological volumes at high resolution. This challenge can potentially be mitigated by balancing the accuracy and computational cost for different components of the forward model while ensuring robust inference. To guide our steps in this, we perform a sensitivity analysis of SBI for galaxy clustering on various components of the cosmological simulations: gravity model, halo finder, and the galaxy–halo distribution models (halo-occupation distribution, HOD). We infer the $\sigma _8$ and $\Omega _\mathrm{ m}$ using galaxy power spectrum multipoles and the bispectrum monopole assuming a galaxy number density expected from the luminous red galaxies observed using the Dark Energy Spectroscopy Instrument. We find that SBI is insensitive to changing gravity model between N-body simulations and particle mesh simulations. However, changing the halo finder from friends of friends to Rockstar can lead to biased estimate of $\sigma _8$ based on the bispectrum. For galaxy models, training SBI on more complex HOD leads to consistent inference for less complex HOD models, but SBI trained on simpler HOD models fails when applied to analyse data from a more complex HOD model. Based on our results, we discuss the outlook on cosmological simulations with a focus on applying SBI approaches to future galaxy surveys.

Testing the framework of the halo occupation distribution with assembly bias modelling and empirical extensions

ABSTRACT We investigate theoretical systematics caused by the application of the halo occupation distribution (HOD) to the study of galaxy clustering at non-linear scales. To do this, we repeat recent cosmological analyses using extended HOD models based on both the Aemulus and Aemulus $\nu$ simulation suites, allowing for variations in the dark matter halo shape, incompleteness, baryonic effects, and position bias of central galaxies. We fit to the galaxy correlation function including the projected correlation function, redshift-space monopole and quadrupole, and consider how the changes in HOD affect the retrieval of cosmological information. These extensions can be understood as an evaluation of the impact of the secondary bias in the clustering analysis. In the application of BOSS (Baryon Oscillation Spectroscopic Survey) galaxies, these changes do not have a significant impact on the measured linear growth rate. However, we do find weak to mild evidence for some of the effects modelled by the empirical parametrizations adopted. The modelling is able to make the HOD approach more complete in terms of cosmological constraints. We anticipate that the future and better data can provide tighter constraints on the new prescriptions of the HOD model.

Quijote-PNG: Optimizing the Summary Statistics to Measure Primordial Non-Gaussianity

We apply a suite of different estimators to the Quijote-png halo catalogs to find the best approach to constrain Primordial non-Gaussianity (PNG) at nonlinear cosmological scales, up to kmax=0.5hMpc−1 . The set of summary statistics considered in our analysis includes the power spectrum, bispectrum, halo mass function, marked power spectrum, and marked modal bispectrum. Marked statistics are used here for the first time in the context of the PNG study. We perform a Fisher analysis to estimate their cosmological information content, showing substantial improvements when marked observables are added to the analysis. Starting from these summaries, we train deep neural networks to perform likelihood-free inference of cosmological and PNG parameters. We assess the performance of different subsets of summary statistics; in the case of fNLequil , we find that a combination of the power spectrum and a suitable marked power spectrum outperforms the combination of power spectrum and bispectrum, the baseline statistics usually employed in PNG analysis. A minimal pipeline to analyze the statistics we identified can be implemented either with our ML algorithm or via more traditional estimators, if these are deemed more reliable.

Non-linear matter power spectrum modeling in interacting dark energy cosmologies

Understanding the behavior of the matter power spectrum on non-linear scales beyond the ΛCDM model is crucial for accurately predicting the large-scale structure (LSS) of the Universe in non-standard cosmologies. In this work, we present an analysis of the non-linear matter power spectrum within the framework of interacting dark energy-dark matter cosmologies (IDE). We employ N-body simulations and theoretical models to investigate the impact of IDE on these non-linear scales. Beginning with N-body simulations characterized by a fixed parameter space delineated by prior observational research, we adeptly fit the simulated spectra with a simple parametric function, achieving accuracy within 5%. Subsequently, we refine a modified halo model tailored to the IDE cosmology, exhibiting exceptional precision in fitting the simulations down to scales of approximately 1 h/Mpc. To assess the model’s robustness, we conduct a forecast analysis for the Euclid survey, employing our refined model. We find that the coupling parameter ξ will be constrained to σ(ξ)=0.0110. This marks a significant improvement by an order of magnitude compared to any other current observational tests documented in the literature. These primary findings pave the way for a novel preliminary approach, enabling the utilization of IDE models for observational constraints concerning LSS data on non-linear scales.

Phase Change Memory Drift Compensation in Spiking Neural Networks Using a Non-Linear Current Scaling Strategy

The non-ideality aspects of phase change memory (PCM) such as drift and resistance variability can pose significant obstacles in neuromorphic hardware implementations. A unique drift and variability compensation strategy is demonstrated and implemented in an FD-SOI SNN hardware unit composed of embedded phase change memories (ePCMs), current attenuators, and spiking neurons. The effect of drift and variability compensation on inference accuracy is tested on the MNIST dataset to show that our drift and variability mitigation strategy is effective in sustaining its accuracy over time. The variability is reduced by up to 5% while the drift coefficient is reduced by up to 57.8%. The drift is compensated and the SNN classification accuracy is sustained for up to 2 years with intrinsic control-free hardware that tracks the ePCM current over time and consumes less than 30 µW. The results are based on ePCM chip experimental data and pos-layout simulation of a test chip comprising the proposed circuit solution.

Proximity Operators of Perspective Functions with Nonlinear Scaling

Proximity Operators of Perspective Functions with Nonlinear Scaling

The renormalization group for large-scale structure: origin of galaxy stochasticity

The renormalization group equations for large-scale structure (RG-LSS) describe how the bias and stochastic (noise) parameters — both of matter and biased tracers such as galaxies — evolve as a function of the cutoff Λ of the effective field theory. In previous work, we derived the RG-LSS equations for the bias parameters using the Wilson-Polchinski framework. Here, we extend these results to include stochastic contributions, corresponding to terms in the effective action that are higher order in the current J. We derive the general local interaction terms that describe stochasticity at all orders in perturbations,and a closed set of nonlinear RG equations for their coefficients. These imply thata single nonlinear bias term generates all stochastic moments through RG evolution. Further, theevolution is controlled by a different, lower scale than the nonlinear scale. This has implications for the optimal choice of the renormalization scale when comparing the theory with data to obtain cosmological constraints.

Impact of assembly bias on clustering plus weak lensing cosmological analysis

Context. Empirical models of galaxy-halo connection such as the halo occupation distribution (HOD) model have been widely used over the past decades to intensively test perturbative models on quasi-linear scales. However, these models fail to reproduce the galaxygalaxy lensing signal on non-linear scales, over-predicting the observed signal by up to 40%. Aims. With ongoing Stage-IV galaxy surveys such as DESI and Euclid that will measure cosmological parameters at sub-percent precision, it is now crucial to precisely model the galaxy-halo connection in order to accurately estimate the theoretical uncertainties of perturbative models. Methods. This paper compares a standard HOD (based on halo mass only) to an extended HOD that incorporates as additional features galaxy assembly bias and local environmental dependencies on halo occupation. These models were calibrated against the observed clustering and galaxy-galaxy lensing signal of eBOSS luminous red galaxies and emission line galaxies in the range 0.6 &lt; z &lt; 1.1. We performed a combined clustering-lensing cosmological analysis on the simulated galaxy samples of both HODs to quantify the systematic budget of perturbative models. Results. By considering not only the mass of the dark matter halos but also these secondary properties, the extended HOD offers a more comprehensive understanding of the connection between galaxies and their surroundings. In particular, we found that the luminous red galaxies preferentially occupy denser and more anisotropic environments. Our results highlight the importance of considering environmental factors in empirical models with an extended HOD that reproduces the observed signal within 20% on scales below 10 h−1 Mpc. Our cosmological analysis reveals that our perturbative model yields similar constraints regardless of the galaxy population, with a better goodness of fit for the extended HOD. These results suggest that the extended HOD should be used to quantify modelling systematics. This extended framework should also prove useful for forward modelling techniques.

Understanding melodic transitions: A neuro-acoustical study with Indian Classical Music

Different kinds of melodic movements and note-to-note transitions form one of the most important features of music. In this work, we have tried to analyze some of these transitions, within the domain of Instrumental Indian Classical Music. Two most widely used transitions in ICM renditions have been chosen for this pilot study- i). slow, gliding transitions, ii). fast, direct-note transitions Three pairs of melodic portions have been prepared from recorded Sitar renditions of Indian maestros, such that within each pair, one clips was predominantly characterized by slow, gliding inter-note transitions and the other by fast, direct ones. Time of transitions, pitch profiles and certain acoustic features like spectral centroid, entropy, kurtosis, skewness etc. were first compared for these two clip-categories. Next, Electroencephalogram (EEG) experiment was performed on 5 participants, using these two kinds of clips as input signals. Extracted neural signals were analyzed with help of robust non-linear scaling technique of ‘Multifractal Detrended Fluctuation Analysis’. The neural responses corresponding to these two contrasting transitions were then compared with parameters like EEG-complexity and EEG-entropy. This pilot-study is a novel attempt to understand the acoustic attributes and characteristics of two contrasting melodic transitions of ICM, as well as their corresponding neural correlates.

FDP-FL: differentially private federated learning with flexible privacy budget allocation

Abstract Federated learning (FL) as a privacy-preserving technology enables multiple clients to collaboratively train models on decentralized data. However, transmitting model parameters between local clients and the central server can potentially result in information leakage. Differentially private federated learning (DPFL) has emerged as a promising solution to enhance privacy. Nevertheless, existing DPFL schemes suffer from two issues: (i) most schemes that aim to achieve desired model accuracy may incur a high privacy budget. (ii) several schemes that consider the trade-off between privacy and accuracy by utilizing linear clipping bound may distort numerous model parameters. In this paper, we first propose FDP-FL, a flexible differential privacy approach for FL. FDP-FL introduces a novel series sum privacy budget allocation instead of uniform allocation and enables adaptive and nonlinear noise scale decay. In this way, a tight bound for cumulative privacy loss can be achieved while optimizing model accuracy. Then in order to mitigate gradient leakages caused by honest-but-curious clients and server, we further design client-level FDP-FL and record-level FDP-FL, respectively. Experimental results demonstrate that our FDP-FL improves model accuracy by $\sim $13.3% compared with the basic DP-FL under a fixed privacy budget and outperforms existing trade-off schemes with the same hyperparameter setting.