Online model error correction with neural networks: application to the Integrated Forecasting System

In recent years, there has been significant progress in the development of fully data-driven global numerical weather prediction models. These machine learning weather prediction models have their strength, notably accuracy and low computational requirements, but also their weakness: they struggle to represent fundamental dynamical balances, and they are far from being suitable for data assimilation experiments. Hybrid modelling emerges as a promising approach to address these limitations. Hybrid models integrate a physics-based core component with a statistical component, typically a neural network, to enhance prediction capabilities. In this article, we propose to develop a model error correction for the operational Integrated Forecasting System (IFS) of the European Centre for Medium-Range Weather Forecasts using a neural network. The neural network is initially pre-trained offline using a large dataset of operational analyses and analysis increments. Subsequently, the trained network is integrated into the IFS within the Object-Oriented Prediction System (OOPS) so as to be used in data assimilation and forecast experiments. It is then further trained online using a recently developed variant of weak-constraint 4D-Var. The results show that the pre-trained neural network already provides a reliable model error correction, which translates into reduced forecast errors in many conditions and that the online training further improves the accuracy of the hybrid model in many conditions.

Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review

Data Assimilation (DA) and Uncertainty quantification (UQ) are extensively used in analysing and reducing error propagation in high-dimensional spatial-temporal dynamics. Typical applications span from computational fluid dynamics (CFD) to geoscience and climate systems. Recently, much effort has been given in combining DA, UQ and machine learning (ML) techniques. These research efforts seek to address some critical challenges in high-dimensional dynamical systems, including but not limited to dynamical system identification, reduced order surrogate modelling, error covariance specification and model error correction. A large number of developed techniques and methodologies exhibit a broad applicability across numerous domains, resulting in the necessity for a comprehensive guide. This paper provides the first overview of the state-of-the-art researches in this interdisciplinary field, covering a wide range of applications. This review aims at ML scientists who attempt to apply DA and UQ techniques to improve the accuracy and the interpretability of their models, but also at DA and UQ experts who intend to integrate cutting-edge ML approaches to their systems. Therefore, this article has a special focus on how ML methods can overcome the existing limits of DA and UQ, and vice versa. Some exciting perspectives of this rapidly developing research field are also discussed.

Online model error correction with neural networks in the incremental 4D-Var framework

Recent studies have demonstrated that it is possible to combine machine learning with data assimilation to reconstruct the dynamics of a physical model partially and imperfectly observed. Data assimilation is used to estimate the system state from the observations, while machine learning computes a surrogate model of the dynamical system based on those estimated states. The surrogate model can be defined as an hybrid combination where a physical model based on prior knowledge is enhanced with a statistical model estimated by a neural network. The training of the neural network is typically done offline, once a large enough dataset of model state estimates is available. By contrast, with online approaches the surrogate model is improved each time a new system state estimate is computed. Online approaches naturally fit the sequential framework encountered in geosciences where new observations become available with time. In a recent methodology paper, we have developed a new weak-constraint 4D-Var formulation which can be used to train a neural network for online model error correction. In the present article, we develop a simplified version of that method, in the incremental 4D-Var framework adopted by most operational weather centres. The simplified method is implemented in the ECMWF Object-Oriented Prediction System, with the help of a newly developed Fortran neural network library, and tested with a two-layer two-dimensional quasi geostrophic model. The results confirm that online learning is effective and yields a more accurate model error correction than offline learning. Finally, the simplified method is compatible with future applications to state-of-the-art models such as the ECMWF Integrated Forecasting System.

Voltage-Dependent Synaptic Plasticity (VDSP): Unsupervised probabilistic Hebbian plasticity rule based on neurons membrane potential

This study proposes voltage-dependent-synaptic plasticity (VDSP), a novel brain-inspired unsupervised local learning rule for the online implementation of Hebb's plasticity mechanism on neuromorphic hardware. The proposed VDSP learning rule updates the synaptic conductance on the spike of the postsynaptic neuron only, which reduces by a factor of two the number of updates with respect to standard spike-timing-dependent plasticity (STDP). This update is dependent on the membrane potential of the presynaptic neuron, which is readily available as part of neuron implementation and hence does not require additional memory for storage. Moreover, the update is also regularized on synaptic weight and prevents explosion or vanishing of weights on repeated stimulation. Rigorous mathematical analysis is performed to draw an equivalence between VDSP and STDP. To validate the system-level performance of VDSP, we train a single-layer spiking neural network (SNN) for the recognition of handwritten digits. We report 85.01 $ \pm $ 0.76% (Mean $ \pm $ S.D.) accuracy for a network of 100 output neurons on the MNIST dataset. The performance improves when scaling the network size (89.93 $ \pm $ 0.41% for 400 output neurons, 90.56 $ \pm $ 0.27 for 500 neurons), which validates the applicability of the proposed learning rule for large-scale computer vision tasks. Interestingly, the learning rule better adapts than STDP to the frequency of input signal and does not require hand-tuning of hyperparameters.

State, global and local parameter estimation using local ensemble Kalman filters: applications to online machine learning of chaotic dynamics

In a recent methodological paper, we have shown how to learn chaotic dynamics along with the state trajectory from sequentially acquired observations, using local ensemble Kalman filters. Here, we more systematically investigate the possibilty to use a local ensemble Kalman filter with either covariance localization or local domains, in order to retrieve the state and a mix of key global and local parameters. Global parameters are meant to represent the surrogate dynamics, for instance through a neural network, which is reminiscent of data-driven machine learning of dynamics, while the local parameters typically stand for the forcings of the model. A family of algorithms for covariance and local domain localization is proposed in this joint state and parameter filter context. In particular, we show how to rigorously update global parameters using a local domain EnKF such as the LETKF, an inherently local method. The approach is tested with success on the 40-variable Lorenz model using several of the local EnKF flavors. A two-dimensional illustration based on a multi-layer Lorenz model is finally provided. It uses radiance-like non-local observations, and both local domains and covariance localization in order to learn the chaotic dynamics, the local forcings, and the couplings between layers. This paper more generally addresses the key question of online estimation of both global and local model parameters.

A comparison of combined data assimilation and machine learning methods for offline and online model error correction

Recent studies have shown that it is possible to combine machine learning methods with data assimilation to reconstruct a dynamical system using only sparse and noisy observations of that system. The same approach can be used to correct the error of a knowledge-based model. The resulting surrogate model is hybrid, with a statistical part supplementing a physical part. In practice, the correction can be added as an integrated term (i.e. in the model resolvent) or directly inside the tendencies of the physical model. The resolvent correction is easy to implement. The tendency correction is more technical, in particular it requires the adjoint of the physical model, but also more flexible. We use the two-scale Lorenz model to compare the two methods. The accuracy in long-range forecast experiments is somewhat similar between the surrogate models using the resolvent correction and the tendency correction. By contrast, the surrogate models using the tendency correction significantly outperform the surrogate models using the resolvent correction in data assimilation experiments. Finally, we show that the tendency correction opens the possibility to make online model error correction, i.e. improving the model progressively as new observations become available. The resulting algorithm can be seen as a new formulation of weak-constraint 4D-Var. We compare online and offline learning using the same framework with the two-scale Lorenz system, and show that with online learning, it is possible to extract all the information from sparse and noisy observations.

Model of the Weak Reset Process in HfOx Resistive Memory for Deep Learning Frameworks

The implementation of current deep learning training algorithms is power-hungry, owing to data transfer between memory and logic units. Oxide-based RRAMs are outstanding candidates to implement in-memory computing, which is less power-intensive. Their weak RESET regime, is particularly attractive for learning, as it allows tuning the resistance of the devices with remarkable endurance. However, the resistive change behavior in this regime suffers many fluctuations and is particularly challenging to model, especially in a way compatible with tools used for simulating deep learning. In this work, we present a model of the weak RESET process in hafnium oxide RRAM and integrate this model within the PyTorch deep learning framework. Validated on experiments on a hybrid CMOS/RRAM technology, our model reproduces both the noisy progressive behavior and the device-to-device (D2D) variability. We use this tool to train Binarized Neural Networks for the MNIST handwritten digit recognition task and the CIFAR-10 object classification task. We simulate our model with and without various aspects of device imperfections to understand their impact on the training process and identify that the D2D variability is the most detrimental aspect. The framework can be used in the same manner for other types of memories to identify the device imperfections that cause the most degradation, which can, in turn, be used to optimize the devices to reduce the impact of these imperfections.

Using machine learning to correct model error in data assimilation and forecast applications

The idea of using machine learning (ML) methods to reconstruct the dynamics of a system is the topic of recent studies in the geosciences, in which the key output is a surrogate model meant to emulate the dynamical model. In order to treat sparse and noisy observations in a rigorous way, ML can be combined to data assimilation (DA). This yields a class of iterative methods in which, at each iteration a DA step assimilates the observations, and alternates with a ML step to learn the underlying dynamics of the DA analysis. In this article, we propose to use this method to correct the error of an existent, knowledge-based model. In practice, the resulting surrogate model is an hybrid model between the original (knowledge-based) model and the ML model. We demonstrate numerically the feasibility of the method using a two-layer, two-dimensional quasi-geostrophic channel model. Model error is introduced by the means of perturbed parameters. The DA step is performed using the strong-constraint 4D-Var algorithm, while the ML step is performed using deep learning tools. The ML models are able to learn a substantial part of the model error and the resulting hybrid surrogate models produce better short- to mid-range forecasts. Furthermore, using the hybrid surrogate models for DA yields a significantly better analysis than using the original model.

Combining data assimilation and machine learning to infer unresolved scale parametrisation

In recent years, machine learning (ML) has been proposed to devise data-driven parametrisations of unresolved processes in dynamical numerical models. In most cases, the ML training leverages high-resolution simulations to provide a dense, noiseless target state. Our goal is to go beyond the use of high-resolution simulations and train ML-based parametrisation using direct data, in the realistic scenario of noisy and sparse observations. The algorithm proposed in this work is a two-step process. First, data assimilation (DA) techniques are applied to estimate the full state of the system from a truncated model. The unresolved part of the truncated model is viewed as a model error in the DA system. In a second step, ML is used to emulate the unresolved part, a predictor of model error given the state of the system. Finally, the ML-based parametrisation model is added to the physical core truncated model to produce a hybrid model.