Simulation-based inference as a paradigm for scientific machine learning in the cryosphere and beyond

Reconstructing spatiotemporal systems from sparse observations remains a long-standing challenge in several domains, including geoscience, air pollution and fluid dynamics. While various data assimilation (DA) and machine learning (ML) methods have shown potential, they still face significant challenges (see [1]): The computational burden of conventional DA algorithms (including error covariance specification), particularly for multivariate, high-dimensional systems. Sparse and movable sensor placement, which makes conventional ML models (typically requiring fixed and regularly distributed input data) cumbersome. The ill-defined nature of the sparse reconstruction problem, which poses significant risks of overfitting. We will present our recent works aimed at addressing these challenges. More specifically, we developed latent DA algorithms [2] to reduce the computational burden of variational DA methods. These algorithms demonstrate great potential in efficiently assimilating sparse observations within a reduced-order latent space constructed by neural networks, thanks to the TorchDA library [3]. The latter enables GPU implementation of mainstream data assimilation methods and supports non-explicit state-observation transformation functions, provided they can be learned by a neural network.We have also employed advanced deep learning techniques, including Voronoi-tessellation CNNs [4] and Vision Transformer-based autoencoders [5], to learn mappings from sparse observations to the complete physical space. These approaches effectively address challenges such as movable sensor placements and varying sensor numbers. Their integration with DA algorithms has also been evaluated.Finally, our recent work explores [6] the utility of generative AI techniques, particularly denoising diffusion models, for field reconstruction from sparse observations. Generative AI methods offer two main advantages: first, they produce a sample from a probability distribution rather than predicting the mean as a fixed output, which can help mitigate overfitting caused by the illy-defined problem. Second, they inherently function as ensemble predictors by generating several samples, facilitating uncertainty quantification, which is essential in data assimilation. The numerical results tested on cases ranging from fluid dynamics benchmarks to semi-operational air pollution simulations will also be discussed.[1] Cheng, S., Quilodrán-Casas, C., Ouala, S., Farchi, A., Liu, C., Tandeo, P., Fablet, R., Lucor, D., Iooss, B., Brajard, J., Xiao, D., Janjic, T., Ding, W., Guo, Y., Carrassi, A., Bocquet, M. and Arcucci, R, 2023. Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review. IEEE/CAA Journal of Automatica Sinica[2] Cheng, S., Chen, J., Anastasiou, C., Angeli, P., Matar, O.K., Guo, Y.K., Pain, C.C. and Arcucci, R., 2023. Generalised latent assimilation in heterogeneous reduced spaces with machine learning surrogate models. Journal of Scientific Computing[3] Cheng, S., Min, J., Liu, C. and Arcucci, R., 2025. TorchDA: A Python package for performing data assimilation with deep learning forward and transformation functions. Computer Physics Communications[4] Cheng, S., Liu, C., Guo, Y. and Arcucci, R., 2024. Efficient deep data assimilation with sparse observations and time-varying sensors. Journal of Computational Physics[5] Fan, H., Cheng, S., de Nazelle, A.J. and Arcucci, R., 2024. ViTAE-SL: a vision transformer-based autoencoder and spatial interpolation learner for field reconstruction. Computer Physics Communications [6] Zhuang, Y., Cheng, S. and Duraisamy, K., 2025. Spatially-aware diffusion models with cross-attention for global field reconstruction with sparse observations. Computer Methods in Applied Mechanics and Engineering

Quantifying continental-scale river discharge is essential to understanding the terrestrial water cycle but is susceptible to errors caused by a lack of observations and the limitations of hydrodynamic modeling. Data assimilation (DA) methods are increasingly used to estimate river discharge in combination with emerging river-related remote sensing products (e.g., water surface elevation [WSE], water surface slope, river width, and flood extent). However, directly comparing simulated WSE to satellite altimetry data remains challenging (e.g., because of large biases between simulations and observations or uncertainties in parameters), and large errors can be introduced when satellite observations are assimilated into hydrodynamic models. In this study we performed direct, anomaly, and normalized value assimilation experiments to investigate the capacity of DA to improve river discharge within the current limitations of hydrodynamic modeling. We performed hydrological DA using a physically-based empirical localization method applied to the Amazon Basin. We used satellite altimetry data from ENVISAT, Jason 1, and Jason 2. Direct DA was the baseline assimilation method and was subject to errors due to biases in the simulated WSE. To overcome these errors, we used anomaly DA as an alternative to direct DA. We found that the modeled and observed WSE distributions differed considerably (e.g., differences in amplitude, seasonal flow variation, and a skewed distribution due to limitations of the hydrodynamic models). Therefore, normalized value DA was performed to improve discharge estimation. River discharge estimates were improved at 24 %, 38 %, and 62 % of stream gauges in the direct, anomaly, and normalized value assimilations relative to simulations without DA. Normalized value assimilation performed best for estimating river discharge given the current limitations of hydrodynamic models. Most gauges within the river reaches covered by satellite observations accurately estimated river discharge, with Nash-Sutcliffe efficiency (NSE) > 0.6. The amplitudes of WSE variation were improved in the normalized DA experiment. Furthermore, in the Amazon Basin, normalized assimilation (median NSE = 0.50) improved river discharge estimation compared to open-loop simulation with the global hydrodynamic model (median NSE = 0.42). River discharge estimation using direct DA methods was improved by 7 % with calibration of river bathymetry based on NSE. The direct DA approach outperformed the other DA approaches when runoff was considerably biased, but anomaly DA performed best when the river bathymetry was erroneous. The uncertainties in hydrodynamic modeling (e.g., river bottom elevation, river width, simplified floodplain dynamics, and the rectangular cross-section assumption) should be improved to fully realize the advantages of river discharge DA through the assimilation of satellite altimetry. This study contributes to the development of a global river discharge reanalysis product that is consistent spatially and temporally.

Ensuring a reliable water supply in the face of changing conditions and growing demand is a critical global challenge. Water distribution networks (WDNs) are essential infrastructure, but traditional modeling methods based on historical data often struggle to adapt in real-time and integrate new information. In order to lower model errors in WDN models, this study explores the use of a Data Assimilation (DA) method that makes use of the Ensemble Kalman Filter. The study explores the effectiveness of the DA method, and a novel Greedy Algorithm (GA) for optimizing the location of sensors. The study shows that the DA method improves the WDN model and the GA is able to successfully determine optimal sensor locations. It was observed that increasing the number of sensors in the WDN increased the effectiveness of the DA method. This study highlights the potential of data assimilation to improve WDN modeling. Water utilities can gain from more precise predictions, greater system performance, and efficient decision-making for the management and maintenance of water distribution networks by allowing models to adapt and improve dynamically

In this study, an innovative MODIS fractional snow cover (SCF) data assimilation (DA) prototype framework that invokes machine learning (ML) techniques and Common land model (CoLM) is proposed to improve the estimation of the snow depth (SD) and the SCF. To validate our new framework, we analyzed two snow seasons from 2013 to 2015 at 46 stations in Northern Xinjiang in China. We developed 12 SCF DA schemes that represent different DA methods (direct insertion (DI) and Ensemble Kalman Filter (EnKF)), observational data (original data and gap‐filled MODIS SCF data), and observation operators (five new snow depletion curves (SDCs) defined using traditional multivariate nonlinear regression and four ML methods). While improving the frequency of the SCF observations in the DI‐based DA scheme only resulted in a marginal improvement in the snow estimates, by adding new SDCs fitted by ML techniques (e.g., deep belief network), and the gap‐filled MODIS SCF data to the EnKF‐based DA scheme, we were able to reduce model structural uncertainties of CoLM and achieve marked improvement in the accuracy of the snow estimates (RMSE = 5.92 cm, mean bias error = −1.94 cm, and average degree of improvement = 32.18% for SD estimates and RMSE = 15. 79%, mean bias error = −1.21%, and average degree of improvement = 47.95% for SCF estimates). Our results demonstrate the feasibility of improving snow estimates by combining the ML techniques with physically based snowpack model in a SCF DA framework.

Challenges for data assimilation – from convective‐scale to climate

Biological and medical diagnoses depend on high-quality measurements. A wearable device based on Internet of Things (IoT) must be unobtrusive to the human body to encourage users to accept continuous monitoring. However, unobtrusive IoT devices are usually of low quality and unreliable because of the limitation of technology progress that has slowed down at high peak. Therefore, advanced inference techniques must be developed to address the limitations of IoT devices. This review proposes that IoT technology in biological and medical applications should be based on a new data assimilation process that fuses multiple data scales from several sources to provide diagnoses. Moreover, the required technologies are ready to support the desired disease diagnosis levels, such as hypothesis test, multiple evidence fusion, machine learning, data assimilation, and systems biology. Furthermore, cross-disciplinary integration has emerged with advancements in IoT. For example, the multiscale modeling of systems biology from proteins and cells to organs integrates current developments in biology, medicine, mathematics, engineering, artificial intelligence, and semiconductor technologies. Based on the monitoring objectives of IoT devices, researchers have gradually developed ambulant, wearable, noninvasive, unobtrusive, low-cost, and pervasive monitoring devices with data assimilation methods that can overcome the limitations of devices in terms of quality measurement. In the future, the novel features of data assimilation in systems biology and ubiquitous sensory development can describe patients’ physical conditions based on few but long-term measurements.

ABSTRACTAn original data assimilation (DA) scheme with a general dynamics model is considered. It is shown that this scheme can be approximated by the stochastic diffusion process. The sufficient conditions to provide this approximation are formulated. Based on this algorithm a new DA method is developed. The method combines variational and statistical approaches commonly used in DA theory and minimizes the variance of the trajectory of a diffusion process in conjunction with a dynamics numerical model. In this sense the method is optimal in contrast to other DA approaches. The proposed scheme takes the model dynamics into account and in this way it differs from the well-known Kalman filter. Furthermore, the derived DA method can be applied to a very wide field of dynamical systems, for example, gas dynamics, fluid dynamics and other disciplines. However, the current study deals with oceanography and DA in oceanography specifically. Then the method is applied to the HYbrid Coordinate Ocean Model and assimilates satellite sea level anomaly data from the Archiving, Validating and Interpolating Satellite Oceanography Data over the Atlantic Ocean to correct the model state. Several numerical experiments have been performed. The experiments show that the method substantially changes the synoptic and mesoscale structure of ocean dynamics. Also, the distribution of the obtained result is estimated through the solution of the Fokker–Planck–Kolmogorov equation.

Ensemble data assimilation methods for improving river water quality forecasting accuracy

Surface observations can provide crucial information for NWP models. If not assimilated carefully, however, they can degrade forecast accuracy, especially in complex terrains like the Alps. The horizontal and vertical covariances of climatological background error covariances used in the three-dimensional variational (3DVar) data assimilation (DA) method can produce unrealistic increments over sloped terrain. For instance, an observation from a valley station can still generate increments at the mountaintop, even though the valley observation may not accurately represent the mountaintop's weather conditions. We used a hybrid three-dimensional ensemble variational (Hybrid-3DEnVar) DA method to address this issue, incorporating a 50-member convection-permitting ensemble. This method was recently tested in Geosphere Austria's convective scale limited-area NWP model AROME at a 2.5 km horizontal resolution. We assimilated 2-meter temperature, 2-meter relative humidity, geopotential, and 10-meter wind components from 680 surface stations, including the Austrian TAWES network and SYNOP observations from neighbouring countries. 400 stations were actively assimilated from the observation dataset, and the rest were used to verify the analysis.  Our results present the effectiveness of this newly tested Hybrid-3DEnVar against GeoSphere Austria's operational 3DVar in assimilating surface observations over complex Alpine terrain.

SummaryEstimating river discharge from in situ and/or remote sensing data is a key issue for evaluation of water balance at local and global scales and for water management. Variational data assimilation (DA) is a powerful approach used in operational weather and ocean forecasting, which can also be used in this context. A distinctive feature of the river discharge estimation problem is the likely presence of significant uncertainty in principal parameters of a hydraulic model, such as bathymetry and friction, which have to be included into the control vector alongside the discharge. However, the conventional variational DA method being used for solving such extended problems often fails. This happens because the control vector iterates (i.e., approximations arising in the course of minimization) result into hydraulic states not supported by the model. In this paper, we suggest a novel version of the variational DA method specially designed for solving estimation‐under‐uncertainty problems, which is based on the ideas of iterative regularization.The method is implemented with SIC2, which is a full Saint‐Venant based 1D‐network model. The SIC2 software is widely used by research, consultant and industrial communities for modeling river, irrigation canal, and drainage network behavior. The adjoint model required for variational DA is obtained by means of automatic differentiation. This is likely to be the first stable consistent adjoint of the 1D‐network model of a commercial status in existence.The DA problems considered in this paper are offtake/tributary estimation under uncertainty in the cross‐device parameters and inflow discharge estimation under uncertainty in the bathymetry defining parameters and the friction coefficient. Numerical tests have been designed to understand identifiability of discharge given uncertainty in bathymetry and friction. The developed methodology, and software seems useful in the context of the future Surface Water and Ocean Topography satellite mission. Copyright © 2016 John Wiley & Sons, Ltd.

Assessment of long-term multisource surface and subsurface soil moisture products and estimate methods on the Tibetan Plateau

We apply information theory within an ensemble‐based data assimilation approach and define information matrix in ensemble subspace. The information matrix in ensemble subspace employs a flow‐dependent forecast error covariance and it is of relatively small dimensions (equal to the ensemble size). The information matrix in ensemble subspace can be directly linked to the information matrix typically used in non‐ensemble‐based data assimilation methods, such as the Kalman Filter (KF) and the three‐dimensional variational (3D‐Var) methods, which provides a framework for consistent comparisons of information measures between different data assimilation methods.We evaluate information measures, such as degrees of freedom for signal, within the Maximum Likelihood Ensemble Filter (MLEF) data assimilation approach and compare them with those obtained using the KF approach and the 3D‐Var approach. We assimilate model‐simulated observations and use the Goddard Earth Observing System Single Column Model (GEOS‐5 SCM) as a dynamical forecast model.The experimental results demonstrate that the proposed framework is useful for comparing information measures obtained in different data assimilation approaches. These comparisons indicate that using a flow‐dependent forecast error covariance matrix (e.g. as in the KF and the MLEF experiments) is fundamentally important for adequately describing prior knowledge about the true model state when calculating information measures of assimilated observations. We also demonstrate that data assimilation results obtained using the KF and the MLEF approach (when ensemble size is larger than 10 ensemble members) are superior to the results of the 3D‐Var approach. Copyright © 2007 Royal Meteorological Society

Data Assimilation for Marine Ecosystem Models

<p>Mesoscale and sub-mesoscale features such as eddies, fronts, boundary currents and localised upwellings play an important role in ocean dynamics, particularly in the coastal / shelf seas which are characterised by the values of Rossby radius much smaller than in the open ocean. Fine resolution ocean modelling is a ubiquitous practice to resolve such features. Due to inevitable errors,ocean models tend to drift from reality. A process called data assimilation (DA) is a way of keeping a model ‘on the tracks’ by constantly correcting it with fresh observations (DARC, 2021). Due to their complexity, operational data assimilation systems, such as NEMOVAR, DART or PDAF remained a topic mostly reserved to experts (Carrassi et al, 2018). This study presents a simple and computationally efficient method for DA in a high-resolution 3D ocean model, which is nested into a coarse-resolution good quality data assimilating (parent) model. The method is particularly suitable for localised coastal models which are run by small modelling groups. The method named Data Assimilation with Stochastic-Deterministic Downscaling (SDDA) reduces bias and root mean square errors (RMSE) of the high-resolution model. The basic idea is to assimilate data from the parent model instead of actual observations. In this way, the high-resolution model is physically aware of observations via the parent model. The method allows to avoid a complex process of assimilating the same observations which were already assimilated into the parent model. The method consists of two stages: (1) downscaling the parent model output onto the child model grid using Stochastic-Deterministic Downscaling (Shapiro et al, 2021), and (2) applying a Kalman gain formula to each of the fine grid nodes. The method is illustrated in a synthetic 2D case where the true solution is known, and the high-resolution model forecast (before data assimilation) is simulated by adding various types of errors. The SDDA method reduces the child model bias to the same level as in the parent model and reduces the RMSE typically by a factor of 2 to 5.</p><p>References</p><p>DARC, 2021 available at https://research.reading.ac.uk/met-darc/aboutus/what-is-data-assimilation/ last access: 05 August 2021</p><p>Carrassi, A., Bocquety ,M., Bertino, L. and Evensen,G. 2018. Data assimilation in the geosciences: An overview of methods, issues, and perspectives. WIRES climate change, v9, No5.</p><p>Shapiro, G. I., Gonzalez-Ondina, J. M., and Belokopytov, V. N., 2021. High-resolution stochastic downscaling method for ocean forecasting models and its application to the Red Sea dynamics, Ocean Sci., 17, 891–907.</p>

Ensemble simulations have been performed with a climate model constrained to follow temperature histories obtained from a recent compilation of 56 well‐calibrated surface temperature proxy records, using a new data assimilation technique. First, we demonstrate that the data assimilation technique provides a faithful representation in the Northern Hemisphere of the signal recorded by the climate proxies at both the regional and gridbox scales. Second, by varying the external forcing, the parameters of the data assimilation method, and the parameters controlling the equilibrium climate sensitivity of the climate model, we demonstrate that the uncertainty in model results is much lower in simulations using data assimilation than in those without it. This observation implies that the data assimilation, using a set of 56 proxies, is providing an efficient and robust constraint on the simulated climate variability over the past centuries. At the hemispheric and continental scales, the model reconstructions using data assimilation are in good agreement with both the instrumental record of the past 150 years and reconstructions of climate in past centuries derived from the application of traditional statistical approaches to networks of proxy data. This increases the confidence in both the data assimilation and traditional statistical approaches. Our data assimilation method, however, is unable to provide a reliable reconstruction over the North Atlantic Ocean, which we attribute to the paucity of proxy data in that region.