Learning-based models for fluid dynamics often operate in unconstrained function spaces, leading to physically inadmissible, unstable simulations. While penalty-based methods offer soft regularization, they provide no structural guarantees, resulting in spurious divergence and long-term collapse. In this work, we introduce a unified framework that enforces the incompressible continuity equation as a hard, intrinsic constraint for both deterministic and generative modeling. First, to project deterministic models onto the divergence-free subspace, we integrate a differentiable spectral Leray projection grounded in the Helmholtz-Hodge decomposition, which restricts the regression hypothesis space to physically admissible velocity fields. Second, to generate physically consistent distributions, we show that simply projecting model outputs is insufficient when the prior is incompatible. To address this, we construct a divergence-free Gaussian reference measure via a curl-based pushforward, ensuring the entire probability flow remains subspace-consistent by construction. Experiments on 2D Navier-Stokes equations demonstrate exact incompressibility up to discretization error and substantially improved stability and physical consistency.

Fine-scale near-surface wind field prediction is essential for a wide range of applications. However, most operational and AI-based weather models operate at kilometer-scale resolution, where terrain-induced wind features such as slope jets, flow deflection, and recirculation are systematically averaged out. Here we introduce FuXi-CFD, a machine learning-based framework designed to generate detailed three-dimensional (3D) near-surface wind fields at 30-meter horizontal resolution, using only coarse-resolution atmospheric inputs and high-resolution terrain information. The model is trained on a large-scale dataset generated via computational fluid dynamics (CFD), encompassing a wide range of terrain types and inflow conditions. Although relying only on horizontal wind inputs, FuXi-CFD infers the full 3D wind fields-including latent variables such as vertical velocity and turbulence-related features. It achieves CFD-comparable accuracy while reducing inference time from hours to seconds. Notably, the model also generalizes well to real-world conditions, as demonstrated by consistent performance against independent wind-tower observations. This capability enables real-time wind field reconstruction for terrain-sensitive applications such as wind turbine siting, power forecasting, and wildfire spread modeling.

Velocity measurement techniques, such as particle image velocimetry (PIV), face a trade-off between field of view, spatial resolution and sampling rate, so that small-scale vortices, shear layers and high-frequency turbulent motions are often under-resolved. Most physics-informed reconstructions use a velocity-pressure formulation, even though pressure is not measured in typical PIV experiments, so the Navier-Stokes constraints are only weakly enforced. We address this issue by formulating a vorticity-velocity physics-informed network (VVPINN), in which pressure is eliminated and incompressibility is enforced together with a vorticity transport equation, thereby directly constraining the velocity field and its derivatives. We then compare this formulation with a conventional velocity-pressure PINN (VPPINN) for spatio-temporal super-resolution of planar PIV data in three cases: a laminar multi-cylinder wake, a two-dimensional Taylor-Green vortex and an experimental two-cylinder wake. In the Taylor-Green vortex case, with identical architectures and training strategies, the VVPINN yields smaller velocity errors, reduces the $L_2$ errors in vorticity and shear by approximately $10\,\%$ , and the pressure gradient errors by up to approximately $30\,\%$ at moderate super-resolution factors, and produces instantaneous fields with more physically plausible vorticity, shear and fine-scale pressure gradient patterns. Spectral analysis shows that the temporal energy spectrum is recovered accurately, whereas the wavenumber spectra, particularly beyond the Nyquist wavenumber, remain more difficult to match because the training data strongly constrain the time histories at sampled locations, but only indirectly inform the smallest spatial scales. Overall, the results indicate that vorticity-based constraints provide a more effective route to physics-consistent super-resolution of sub-sampled PIV data than the conventional velocity-pressure formulation.

High-resolution wind information is essential for wind energy planning and power forecasting, particularly in regions with complex terrain. However, most AI-based weather forecasting models operate at kilometer-scale resolution, constrained by the reanalysis datasets they are trained on. Here we introduce FuXi-CFD, an AI-based downscaling framework designed to generate detailed three-dimensional wind fields at 30-meter horizontal resolution, using only coarse-resolution atmospheric inputs. The model is trained on a large-scale dataset generated via computational fluid dynamics (CFD), encompassing a wide range of terrain types, surface roughness, and inflow conditions. Remarkably, FuXi-CFD predicts full 3D wind structures – including vertical wind and turbulent kinetic energy – based solely on horizontal wind input at 10 meters above ground, the typical output of AI-based forecast systems. It achieves CFD-comparable accuracy while reducing inference time from hours to seconds. By bridging the resolution gap between regional forecasts and site-specific wind dynamics, FuXi-CFD offers a scalable and operationally efficient solution to support the growing demands of renewable energy deployment.

Intracranial aneurysm (IA) is a common cerebrovascular disease that is usually asymptomatic but may cause severe subarachnoid hemorrhage (SAH) if ruptured. Although clinical practice is usually based on individual factors and morphological features of the aneurysm, its pathophysiology and hemodynamic mechanisms remain controversial. To address the limitations of current research, this study constructed a comprehensive hemodynamic dataset of intracranial aneurysms. The dataset is based on 466 real aneurysm models, and 10,000 synthetic models were generated by resection and deformation operations, including 466 aneurysm-free models and 9,534 deformed aneurysm models. The dataset also provides medical image-like segmentation mask files to support insightful analysis. In addition, the dataset contains hemodynamic data measured at eight steady-state flow rates (0.001 to 0.004 kg/s), including critical parameters such as flow velocity, pressure, and wall shear stress, providing a valuable resource for investigating aneurysm pathogenesis and clinical prediction. This dataset will help advance the understanding of the pathologic features and hemodynamic mechanisms of intracranial aneurysms and support in-depth research in related fields. Dataset hosted at https://github.com/Xigui-Li/Aneumo.

Centrifugal separation is a highly efficient technique for accelerating the sedimentation of blood constituents in a cylindrical container through high-speed spin-up rotation. Few studies have reported on the separation of different blood constituents from homogeneous mixture of whole blood. In this study, the process through which blood constituent sedimentation occurs in a spin-up rotating cylindrical container is numerically investigated. Whole blood is considered a homogeneous mixture of red blood cells (RBC) and plasma, which are both considered incompressible viscous liquids. The Euler multi-fluid VOF (volume of fluid) model is introduced to simulate the separation of RBCs and plasma. The effects of the rotation speed and the geometric construction of the cylindrical container on the sedimentation and stratification of different blood constituents are studied. A stable interface between the RBC layer and plasma layer forms earlier in a high position. With an increase in the rotation speed, the interface between the RBCs and plasma layers forms more quickly. In the cylindrical container with a helical groove on the outer wall, a stable vortex occurs near the groove, which forces red blood cells to move toward the lower location of the groove, resulting in a conical distribution of the RBC layer and a larger volume fraction of plasma near the exit at the top. This allows for sufficient precipitation of the plasma, improving the separation efficiency.

Background Asymmetry in motor dysfunction and associated dopaminergic deficit is a common characteristic of Parkinson's disease (PD), yet potential explanations remain mysterious. Hereby, we assessed whether asymmetry in the nasal cavity is related to dopaminergic dysfunction asymmetry in PD patients. Methods This cross-sectional, multi-center observational study included 761 PD patients from three cohorts. First, we analyzed data from the Huashan Parkinsonian PET Imaging Database (March 2011 to February 2020), which served as the primary cohort (n=333). Second, we collected de novo data from all PD inpatients in the Hongqiao Campus of Huashan Hospital as internal validation cohort (May 2023 to July 2024, n=77). Finally, we used data from the Parkinson's Progression Markers Initiative as an external validation cohort (n=351). All cohorts included imaging data of structural MRI or CT, as well as dopaminergic neuroimaging using 11C−CFT, 18F−FP−CIT, or 18F−DTBZ on PET or 123I−DaTscan on SPECT. Nasal cavity asymmetry was assessed by visually inspecting the position of nasal septum deviation in structural MRI or CT to determine the dominant side. Both qualitative and quantitative analyses were performed to evaluate the correlations between nasal cavity asymmetry and dopaminergic deficit asymmetry. Results In the primary cohort, 70.2% of patients exhibited consistency between the dominant side of the nasal cavity and the side with a more severe dopaminergic deficit in striatum (φ=0.40, p<0.001). The striatal−specific binding ratios of dopamine uptake were significantly lower on the side ipsilateral to the dominant nasal cavity, and a significantly inverse correlation was found between the asymmetry index of nasal cavity surface area and that of the dopaminergic deficit (r=−0.31, p<0.001). Similar patterns were observed in internal (78.6%, φ=0.57, p<0.001) and external validation cohorts (73.1%, φ=0.45, p<0.001). A stronger correlation was found in sporadic PD (φ=0.67, p<0.001) compared to genetic PD patients (φ=0.31, p=0.3). Conclusions We made a novel and robust observation that the nasal cavity asymmetry is correlated with asymmetry in striatal dopaminergic deficiency in PD patients. The finding may have significant implications for both the etiological and clinical research of PD, supporting the nasal pathway as a potential route for both environmental pathogens and PD treatment. ### Competing Interest Statement The authors have declared no competing interest. ### Funding Statement This work received financial support from the Shanghai Municipal Science and Technology Major Project (grants No. 22JC1410402), National Natural Science Foundation of China (grants No. 82171421, 82371432, 92249302, 82272039, 82021002, 81971641, 81902282, 82171252, and 82371266), National Health Commission of China (grants No. Pro20211231084249000238), the Research project of Shanghai Health Commission (grants No. 2020YJZX0111), the Clinical Research Plan of SHDC (grants No. SHDC2020CR1038B), and the STI2030-Major Projects (grants No. 2022ZD0211600). ### Author Declarations I confirm all relevant ethical guidelines have been followed, and any necessary IRB and/or ethics committee approvals have been obtained. Yes The details of the IRB/oversight body that provided approval or exemption for the research described are given below: This study was approved by the Institutional Review Boards of Huashan Hospital. I confirm that all necessary patient/participant consent has been obtained and the appropriate institutional forms have been archived, and that any patient/participant/sample identifiers included were not known to anyone (e.g., hospital staff, patients or participants themselves) outside the research group so cannot be used to identify individuals. Yes I understand that all clinical trials and any other prospective interventional studies must be registered with an ICMJE-approved registry, such as ClinicalTrials.gov. I confirm that any such study reported in the manuscript has been registered and the trial registration ID is provided (note: if posting a prospective study registered retrospectively, please provide a statement in the trial ID field explaining why the study was not registered in advance). Yes I have followed all appropriate research reporting guidelines, such as any relevant EQUATOR Network research reporting checklist(s) and other pertinent material, if applicable. Yes All data produced in the present study are available upon reasonable request to the authors.

Deterministic Lateral Displacement (DLD) is a microfluidic technology that separates particles based on size and deformability, with separation efficiency closely tied to the critical diameter. In recent years, machine learning (ML) has emerged as a powerful tool for predicting the critical diameter, offering a promising alternative to time-consuming simulations and experiments. However, traditional regression ML models relying on control parameters often struggle to achieve high prediction accuracy, particularly when dealing with asymmetric shapes, due to their limited ability to capture geometric intricacies. To overcome these constraints, an image-based method for predicting the critical diameter is proposed in present study, integrating Convolutional Neural Networks (CNN) with Dissipative Particle Dynamics (DPD). This CNN-DPD approach demonstrates superior prediction performance compared to conventional regression models and remains effective even when trained on small datasets. Specifically, it achieves the same prediction accuracy comparable to that of traditional models trained on 3000 samples, while requiring only 300 samples. Building upon this, a CNN-based framework for optimizing the DLD pillar shape is introduced. Using this framework, it is found that an asymmetric shape along the x-axis outperforms other configurations, and the mechanism by which it reduces the critical diameter-by modulating the peak shift of inter-pillar flow velocity-is elucidated through DPD simulations.

Deterministic Lateral Displacement (DLD) has been widely utilized for the high-throughput and efficient separation of microspheres, cells, exosomes, and proteins, playing a crucial role in size-based particle separation. The high performance of DLD devices in various tasks relies on optimal design. However, current DLD design lacks clear guidance and heavily depends on expert experience, requiring extensive repetitive microfluidic experiments and numerical simulations. This leads to significant economic and time costs, while the complexity of instruments and algorithms further raises the barrier for DLD design and optimization. To address this challenge, this paper proposes a novel integrated approach that combines Dissipative Particle Dynamics (DPD) simulation with various machine learning (ML) models to rapidly predict the critical diameter of DLD devices with arbitrary pillar shapes considering fluid dynamics. The study uses control parameters of B & eacute;zier curves to represent the arbitrary pillar shapes, along with the forces in the x and y directions of the fluid as input parameters. The critical diameter serves as output parameter. Four ML models are trained: Random Forest Regression (RF), Extreme Gradient Boosting (XGBoost), Support Vector Regression (SVR) and Artificial Neural Networks (ANN). To address the low interpretability of complex ML models, the Shapley Additive Explanations (SHAP) method is introduced to clarify all input features. The results demonstrate that ML techniques are highly effective in predicting the critical diameter within DLD devices, with the ANN model achieving superior performance, attaining an $ R<^>2 $ R2 value of 0.949. The SHAP analysis reinforces established fluid dynamics principles. Additionally, it reveals the significant influence of asymmetrical pillars on critical diameter.

Cavitation, the formation of vapor bubbles as the liquid pressure is reduced below the saturated vapor pressure, often requires a substantial negative relative pressure in a pure liquid. Classical nucleation theory (CNT) provides an estimate for the rate of cavitation but there is often a disconnect between the predictions at the molecular scale compared to observations at the macroscale. We report on mesoscale simulations of cavitation based on many-body dissipative particle dynamics (mDPD), a coarse-grained molecular dynamics (MD), which bridges the two scales. A liquid layer is confined between smooth planar walls at a constant temperature, while the pressure is reduced slowly by expanding the wall-bounded domain. The wetting properties of the liquid are determined by the parameters of the interaction potentials. With hydrophilic walls, homogeneous nucleation is observed in the liquid bulk. As a bubble forms and grows, it creates a strong pressure pulse and oscillations that cause other bubbles that may have formed slightly later to collapse. For a nearly neutral wall with a contact angle close to 90[Formula: see text], heterogeneous nucleation occurs at the walls at a smaller negative pressure and generates weaker pressure oscillations. With hydrophobic walls or seed particles, heterogeneous nucleation readily occurs, where fluctuations and the merger of transient surface bubbles are significant.

Deterministic lateral displacement (DLD) devices have shown considerable promise in various applications but require further optimization to meet the demands of high-throughput processing and complex sample adaptability. Recent studies utilizing single-task learning (STL) within a machine learning framework have achieved initial success in predicting critical diameters in DLD systems, offering significant gains in efficiency and cost-effectiveness compared to traditional microfluidic experiments and numerical simulations. However, the inability of STL models to effectively capture inter-task relationships and shared physical features limits both prediction accuracy and generalization capability. To address these limitations, this study proposes a multi-task learning (MTL)-based deep learning framework for the simultaneous prediction of key flow characteristics in DLD devices, including inter-pillar flow fields and streamwise velocity profiles along decision lines. The method employs Bezier curves to generate diverse pillar geometries, with high-quality labeled data acquired through dissipative particle dynamics simulations. A shared-feature neural network is constructed to enable joint modeling of multiple flow-related objectives. The results indicate that the proposed MTL model not only achieves high-prediction accuracy but also substantially improves data efficiency and model generalizability. Compared to conventional STL approaches, the MTL framework allows for the parallel inference of multiple tasks within milliseconds using only images of the pillar array structures, without relying on additional physical assumptions or geometric simplifications. Overall, this study presents a MTL model for the intelligent design of DLD devices and extends the applicability of MTL in microscale flow modeling.

Structural abnormalities of the upper airway can lead to various complications, including obstructive sleep apnea (OSA), a condition that is increasingly diagnosed in the general population. To treat OSA, mandibular advancement devices (MADs) have emerged as a highly promising approach. Traditionally, the evaluation of MAD efficacy has been primarily qualitative. The key innovation of this study lies in the novel application of magnetic resonance imaging (MRI) to isolate and reconstruct the inspiratory phase of upper airway geometry for computational fluid dynamics (CFD) analysis, enabling precise characterization of dynamic airflow patterns during the most vulnerable period for airway collapse. In this proof-of-concept study based on a single-case dataset, fluid dynamics analysis is employed to perform quantitative analysis, providing a detailed understanding of the changes in upper airway airflow characteristics before and after the use of MADs in one OSA patient through MRI. Through quantitative assessments, this study enhances our understanding of the mechanisms underlying OSA and the therapeutic efficacy of MADs. Specifically, the findings from this case reveal that MADs can expand the minimum cross-sectional area (CSAmin) of the upper airway by 102.72%, increase the volume of the velopharynx by 61.98% and the oropharynx by 99.61%, while reducing the maximum airflow velocity by 67.09%. Furthermore, significant reductions in pressure drop, maximum viscous forces, and airway resistance were observed, alongside decreased recirculation and reversed flow in the oropharyngeal region. Additionally, MADs increased pressure in the velopharynx and oropharynx, thereby potentially mitigating the risk of upper airway collapse in OSA patients. This method can effectively isolate the inspiratory-phase upper airway model from MRI data. These preliminary results offer novel insights and strategies for the treatment of upper airway disorders from the perspective of CFD.

Inspired by the wriggling motion of mammals after bathing, it has been discovered that centripetal acceleration can effectively remove droplets. Since rotation emerges as a viable method for this purpose, it is essential to understand the interaction between droplets and rotating structured surfaces. In this study, we numerically investigate the dynamic behavior of droplets and the factors influencing droplets' detachment from a rotating brush-like surface with micro-pillars by employing many-body dissipative particle dynamics. We identify three distinct droplet motion modes: remaining on the surface (Scenario I), undergoing pinch-off (Scenario II), and direct detachment within two rotational cycling periods (Scenario III). Furthermore, regardless of the mode, the center-of-mass velocity of the droplet along the z-axis changes synchronously with the angular velocity direction of the brush-like structure. Our results demonstrate that increasing the rotation radius and reducing the rotation period act synergistically to promote droplet detachment. Additionally, two distinct modes of direct detachment are observed. A critical Weber number (We) is identified for the transition from the stretching mode to the peeling mode, governed by the balance between adhesion force and centripetal force. This study offers valuable insights into active water removal and droplet manipulation techniques.

Intracranial aneurysms (IAs) are serious cerebrovascular lesions found in approximately 5% of the general population. Their rupture may lead to high mortality. Current methods for assessing IA risk focus on morphological and patient-specific factors, but the hemodynamic influences on IA development and rupture remain unclear. While accurate for hemodynamic studies, conventional computational fluid dynamics (CFD) methods are computationally intensive, hindering their deployment in large-scale or real-time clinical applications. To address this challenge, we curated a large-scale, high-fidelity aneurysm CFD dataset to facilitate the development of efficient machine learning algorithms for such applications. Based on 427 real aneurysm geometries, we synthesized 10,660 3D shapes via controlled deformation to simulate aneurysm evolution. The authenticity of these synthetic shapes was confirmed by neurosurgeons. CFD computations were performed on each shape under eight steady-state mass flow conditions, generating a total of 85,280 blood flow dynamics data covering key parameters. Furthermore, the dataset includes segmentation masks, which can support tasks that use images, point clouds or other multimodal data as input. Additionally, we introduced a benchmark for estimating flow parameters to assess current modeling methods. This dataset aims to advance aneurysm research and promote data-driven approaches in biofluids, biomedical engineering, and clinical risk assessment. The code and dataset are available at: https://github.com/Xigui-Li/Aneumo.

Asymmetric bifurcate are common in the vascular system, and circulating tumour cells (CTCs) tend to metastasize at these locations. In this study, the dissipative particle dynamics method is used to simulate the motion of a CTC in an asymmetric microvessel combined with a spring-based network model. Effects of flow rate, the presence of RBCs and haematocrit on the motion of the CTC are investigated. The results indicate that under higher flow rate, the CTC could not adhere to the microvessel and invariably migrated towards the branch exhibiting a higher flow rate. In the presence of RBCs, the initially attached CTC would disassociate from the parent vessel wall under the collisions with them, and it finally enters into the large branch. While at lower flow rates, the RBCs pass over the CTC that is rolling along the top wall of the parent vessel and overtake it, facilitating the CTC adhesion, it ultimately moves into the small branch. At a mediate flow rate, though the attached CTC detaches and enters into the large branch, it would touch the inner wall of the branch. With the haematocrit increasing up to 5.1%, the CTC retains rolling along the vessel wall and is ultimately captured in the small branch.

Significant advancements in the development of machine learning (ML) models for weather forecasting have produced remarkable results. State-of-the-art ML-based weather forecast models, such as FuXi, have demonstrated superior statistical forecast performance in comparison to the high-resolution forecasts (HRES) of the European Centre for Medium-Range Weather Forecasts (ECMWF). However, a common limitation of these ML models is their tendency to generate increasingly smooth predictions as forecast lead times increase, which often results in the underestimation of intensities of extreme weather events. To address this challenge, we developed the FuXi-Extreme model, which employs a denoising diffusion probabilistic model (DDPM) to enhance finer-scale details in the surface forecast data generated by the FuXi model in 5-day forecasts. An evaluation of extreme total precipitation (TP), 10-meter wind speed (WS10), and 2-meter temperature (T2M) illustrates the superior performance of FuXi-Extreme over both FuXi and HRES. Moreover, when evaluating tropical cyclone (TC) forecasts based on International Best Track Archive for Climate Stewardship (IBTrACS) dataset, both FuXi and FuXi-Extreme shows superior performance in TC track forecasts compared to HRES, but they show inferior performance in TC intensity forecasts in comparison to HRES.

Skillful subseasonal forecasts are crucial for various sectors of society but pose a grand scientific challenge. Recently, machine learning-based weather forecasting models outperform the most successful numerical weather predictions generated by the European Centre for Medium-Range Weather Forecasts (ECMWF), but have not yet surpassed conventional models at subseasonal timescales. This paper introduces FuXi Subseasonal-to-Seasonal (FuXi-S2S), a machine learning model that provides global daily mean forecasts up to 42 days, encompassing five upper-air atmospheric variables at 13 pressure levels and 11 surface variables. FuXi-S2S, trained on 72 years of daily statistics from ECMWF ERA5 reanalysis data, outperforms the ECMWF's state-of-the-art Subseasonal-to-Seasonal model in ensemble mean and ensemble forecasts for total precipitation and outgoing longwave radiation, notably enhancing global precipitation forecast. The improved performance of FuXi-S2S can be primarily attributed to its superior capability to capture forecast uncertainty and accurately predict the Madden-Julian Oscillation (MJO), extending the skillful MJO prediction from 30 days to 36 days. Moreover, FuXi-S2S not only captures realistic teleconnections associated with the MJO but also emerges as a valuable tool for discovering precursor signals, offering researchers insights and potentially establishing a new paradigm in Earth system science research.

Bifurcated vessels represent a typical vascular unit of the cardiovascular system. In this study, the blood flow in symmetric and asymmetric bifurcated vessels are simulated based on computational fluid dynamics method. The blood is modeled as non-Newtonian fluid, and the pulsatile flow velocity is applied on the inlet. The effects of the fluid model, bifurcation angle and symmetry of the geometry of the vessel are investigated. The results show that the wall shear stress (WSS) on the outer wall of daughter branches for the non-Newtonian fluid flow is greater than that for Newtonian fluid flow, and the discrepancy between the flow of two fluid models is obvious at relatively low flow rates. With the bifurcation angle increases, the peak axial velocity of the cross-section of daughter branch decreases, so the WSS increases. For the non-Newtonian fluid flow in the asymmetric bifurcated vessels, more flow passes through the daughter vessel with a lower angle, and the WSS along the outer wall of which is lower. Furthermore, the region with a low time-averaged wall stress (TAWSS) and high oscillating shear index(OSI) distributed on the outer wall of bifurcation vessels are larger for the flow in the vessel with smaller bifurcation angle. In conclusion, the effects of the blood viscosity cannot be neglected at low flow rates and the geometry of the bifurcated vessel plays a key role with regards to the blood flow.

Deterministic Lateral Displacement (DLD) device has gained widespread recognition and trusted for filtering blood cells. However, there remains a crucial need to explore the complex interplay between deformable cells and flow within the DLD device to improve its design. This paper presents an approach utilizing a mesoscopic cell-level numerical model based on dissipative particle dynamics to effectively capture this complex phenomenon. To establish the model's credibility, a series of numerical simulations were conducted and the numerical results were validated with nominal experimental data from the literature. These include single cell stretching experiment, comparisons of the morphological characteristics of cells in DLD, and comparison the specific row-shift fraction of DLD required to initiate the zigzag mode. Additionally, we investigate the effect of cell rigidity, which serves as an indicator of cell health, on average flow velocity, trajectory, and asphericity. Moreover, we extend the existing theory of predicting zigzag mode for solid spherical particles to encompass the behavior of red blood cells. To achieve this, we introduce a new concept of effective diameter and demonstrate its applicability in providing highly accurate predictions across a wide range of conditions.

Skillful subseasonal forecasts beyond 2 weeks are crucial for a wide range of applications across various sectors of society. Recently, state-of-the-art machine learning based weather forecasting models have made significant advancements, outperforming the high-resolution forecast (HRES) from the European Centre for Medium-Range Weather Forecasts (ECMWF). However, the full potential of machine learning models in subseasonal forecasts has yet to be fully explored. In this study, we introduce FuXi Subseasonal-to-Seasonal (FuXi-S2S), a machine learning based subseasonal forecasting model that provides global daily mean forecasts up to 42 days, covering 5 upper-air atmospheric variables at 13 pressure levels and 11 surface variables. FuXi-S2S integrates an enhanced FuXi base model with a perturbation module for flow-dependent perturbations in hidden features, and incorporates Perlin noise to perturb initial conditions. The model is developed using 72 years of daily statistics from ECMWF ERA5 reanalysis data. When compared to the ECMWF Subseasonal-to-Seasonal (S2S) reforecasts, the FuXi-S2S forecasts demonstrate superior deterministic and ensemble forecasts for total precipitation (TP), outgoing longwave radiation (OLR), and geopotential at 500 hPa (Z500). Although it shows slightly inferior performance in predicting 2-meter temperature (T2M), it has clear advantages over land area. Regarding the extreme forecasts, FuXi-S2S outperforms ECMWF S2S globally for TP. Furthermore, FuXi-S2S forecasts surpass the ECMWF S2S reforecasts in predicting the Madden Julian Oscillation (MJO), a key source of subseasonal predictability. They extend the skillful prediction of MJO from 30 days to 36 days.