A common strategy for the dimensionality reduction of nonlinear partial differential equations (PDEs) relies on the use of the proper orthogonal decomposition (POD) to identify a reduced subspace and the Galerkin projection for evolving dynamics in this reduced space. However, advection-dominated PDEs are represented poorly by this methodology since the process of truncation discards important interactions between higher-order modes during time evolution. In this study, we demonstrate that encoding using convolutional autoencoders (CAEs) followed by a reduced-space time evolution by recurrent neural networks overcomes this limitation effectively. We demonstrate that a truncated system of only two latent space dimensions can reproduce a sharp advecting shock profile for the viscous Burgers equation with very low viscosities, and a six-dimensional latent space can recreate the evolution of the inviscid shallow water equations. Additionally, the proposed framework is extended to a parametric reduced-order model by directly embedding parametric information into the latent space to detect trends in system evolution. Our results show that these advection-dominated systems are more amenable to low-dimensional encoding and time evolution by a CAE and recurrent neural network combination than the POD-Galerkin technique.

Reduced-order modeling of advection-dominated systems with recurrent neural networks and convolutional autoencoders

A common strategy for the dimensionality reduction of nonlinear partial differential equations relies on the use of the proper orthogonal decomposition (POD) to identify a reduced subspace and the Galerkin projection for evolving dynamics in this reduced space. However, advection-dominated PDEs are represented poorly by this methodology since the process of truncation discards important interactions between higher-order modes during time evolution. In this study, we demonstrate that an encoding using convolutional autoencoders (CAEs) followed by a reduced-space time evolution by recurrent neural networks overcomes this limitation effectively. We demonstrate that a truncated system of only two latent-space dimensions can reproduce a sharp advecting shock profile for the viscous Burgers equation with very low viscosities, and a six-dimensional latent space can recreate the evolution of the inviscid shallow water equations. Additionally, the proposed framework is extended to a parametric reduced-order model by directly embedding parametric information into the latent space to detect trends in system evolution. Our results show that these advection-dominated systems are more amenable to low-dimensional encoding and time evolution by a CAE and recurrent neural network combination than the POD Galerkin technique.

In this paper, we propose Deep Data Assimilation (DDA), an integration of Data Assimilation (DA) with Machine Learning (ML). DA is the Bayesian approximation of the true state of some physical system at a given time by combining time-distributed observations with a dynamic model in an optimal way. We use a ML model in order to learn the assimilation process. In particular, a recurrent neural network, trained with the state of the dynamical system and the results of the DA process, is applied for this purpose. At each iteration, we learn a function that accumulates the misfit between the results of the forecasting model and the results of the DA. Subsequently, we compose this function with the dynamic model. This resulting composition is a dynamic model that includes the features of the DA process and that can be used for future prediction without the necessity of the DA. In fact, we prove that the DDA approach implies a reduction of the model error, which decreases at each iteration; this is achieved thanks to the use of DA in the training process. DDA is very useful in that cases when observations are not available for some time steps and DA cannot be applied to reduce the model error. The effectiveness of this method is validated by examples and a sensitivity study. In this paper, the DDA technology is applied to two different applications: the Double integral mass dot system and the Lorenz system. However, the algorithm and numerical methods that are proposed in this work can be applied to other physics problems that involve other equations and/or state variables.

Deep Data Assimilation: Integrating Deep Learning with Data Assimilation

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.

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

A Reduced Order Deep Data Assimilation model

: Parameter identiﬁcation for wildﬁre forecasting models often relies on case-by-case tuning or posterior diagnosis/analysis, which can be computationally expensive due to the complexity of the forward prediction model. In this paper, we introduce an efﬁcient parameter ﬂexible ﬁre prediction algorithm based on machine learning and reduced order modelling techniques. Using a training dataset generated by physics-based ﬁre simulations, the method forecasts burned area at different time steps with a low computational cost. We then address the bottleneck of efﬁcient parameter estimation by developing a novel inverse approach relying on data assimilation techniques (latent assimilation) in the reduced order space. The forward and the inverse modellings are tested on two recent large wildﬁre events in California. Satellite observations are used to validate the forward prediction approach and identify the model parameters. By combining these forward and inverse approaches, the system manages to integrate real-time observations for parameter adjustment, leading to more accurate future predictions.

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Parameter Flexible Wildfire Prediction Using Machine Learning Techniques: Forward and Inverse Modelling

/pdf/urban-air-pollution-forecasts-generated-from-latent-space-rhel0tjzhu.pdf

Urban air pollution forecasts generated from latent space representation

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 surro-gate 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 state-of-the-art researches in this interdisciplinary field, covering a wide range of applications. This review is aimed 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.

Machine Learning With Data Assimilation and Uncertainty Quantification for Dynamical Systems: A Review

There is an urgent need to build models to tackle Indoor Air Quality issue. Since the model should be accurate and fast, Reduced Order Modelling technique is used to reduce the dimensionality of the problem. The accuracy of the model, that represent a dynamic system, is improved integrating real data coming from sensors using Data Assimilation techniques. In this paper, we formulate a new methodology called Latent Assimilation that combines Data Assimilation and Machine Learning. We use a Convolutional neural network to reduce the dimensionality of the problem, a Long-Short-Term-Memory to build a surrogate model of the dynamic system and an Optimal Interpolated Kalman Filter to incorporate real data. Experimental results are provided for CO2 concentration within an indoor space. This methodology can be used for example to predict in real-time the load of virus, such as the SARS-COV-2, in the air by linking it to the concentration of CO2.

César Quilodrán Casas

Papers

A Reduced Order Deep Data Assimilation model

Parameter Flexible Wildfire Prediction Using Machine Learning Techniques: Forward and Inverse Modelling

Urban air pollution forecasts generated from latent space representation

Machine Learning With Data Assimilation and Uncertainty Quantification for Dynamical Systems: A Review

Data Assimilation in the Latent Space of a Neural Network.