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This article presents a general framework for recovering missing dynamical systems using available data and machine learning techniques. The proposed framework reformulates the prediction problem as a supervised learning problem to approximate a map that takes the memories of the resolved and identifiable unresolved variables to the missing components in the resolved dynamics. We demonstrate the effectiveness of the proposed framework with a theoretical guarantee of a path-wise convergence of the resolved variables up to finite time and numerical tests on prototypical models in various scientific domains. These include the 57-mode barotropic stress models with multiscale interactions that mimic the blocked and unblocked patterns observed in the atmosphere, the nonlinear Schr{o}dinger equation which found many applications in physics such as optics and Bose-Einstein-Condense, the Kuramoto-Sivashinsky equation which spatiotemporal chaotic pattern formation models trapped ion mode in plasma and phase dynamics in reaction-diffusion systems. While many machine learning techniques can be used to validate the proposed framework, we found that recurrent neural networks outperform kernel regression methods in terms of recovering the trajectory of the resolved components and the equilibrium one-point and two-point statistics. This superb performance suggests that recurrent neural networks are an effective tool for recovering the missing dynamics that involves approximation of high-dimensional functions.
This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated as a superv
Physics-informed Machine Learning has recently become attractive for learning physical parameters and features from simulation and observation data. However, most existing methods do not ensure that the physics, such as balance laws (e.g., mass, mome
In this paper, we propose a model-operator-data network (MOD-Net) for solving PDEs. A MOD-Net is driven by a model to solve PDEs based on operator representation with regularization from data. In this work, we use a deep neural network to parameteriz
This is the third paper in a series in which we develop machine learning (ML) moment closure models for the radiative transfer equation (RTE). In our previous work cite{huang2021gradient}, we proposed an approach to learn the gradient of the unclosed
Academics and practitioners have studied over the years models for predicting firms bankruptcy, using statistical and machine-learning approaches. An earlier sign that a company has financial difficulties and may eventually bankrupt is going in emph{