Enforcing sparse structure within learning has led to significant advances in the field of data-driven discovery of dynamical systems. However, such methods require access not only to time-series of the state of the dynamical system, but also to the time derivative. In many applications, the data are available only in the form of time-averages such as moments and autocorrelation functions. We propose a sparse learning methodology to discover the vector fields defining a (possibly stochastic or partial) differential equation, using only time-averaged statistics. Such a formulation of sparse learning naturally leads to a nonlinear inverse problem to which we apply the methodology of ensemble Kalman inversion (EKI). EKI is chosen because it may be formulated in terms of the iterative solution of quadratic optimization problems; sparsity is then easily imposed. We then apply the EKI-based sparse learning methodology to various examples governed by stochastic differential equations (a noisy Lorenz 63 system), ordinary differential equations (Lorenz 96 system and coalescence equations), and a partial differential equation (the Kuramoto-Sivashinsky equation). The results demonstrate that time-averaged statistics can be used for data-driven discovery of differential equations using sparse EKI. The proposed sparse learning methodology extends the scope of data-driven discovery of differential equations to previously challenging applications and data-acquisition scenarios.
We introduce a practical method for incorporating equality and inequality constraints in global optimization methods based on stochastic interacting particle systems, specifically consensus-based optimization (CBO) and ensemble Kalman inversion (EKI). Unlike other approaches in the literature, the method we propose does not constrain the dynamics to the feasible region of the state space at all times; the particles evolve in the full space, but are attracted towards the feasible set by means of a penalization term added to the objective function and, in the case of CBO, an additional relaxatio…
The Ensemble Kalman Filter (EnKF) belongs to the class of iterative particle filtering methods and can be used for solving control--to--observable inverse problems. In this context, the EnKF is known as Ensemble Kalman Inversion (EKI). In recent years several continuous limits in the number of iteration and particles have been performed in order to study properties of the method. In particular, a one--dimensional linear stability analysis reveals possible drawbacks in the phase space of moments provided by the continuous limits of the EKI, but observed also in the multi--dimensional setting. I…
The Bayesian approach to inverse problems is widely used in practice to infer unknown parameters from noisy observations. In this framework, the ensemble Kalman inversion has been successfully applied for the quantification of uncertainties in various areas of applications. In recent years, a complete analysis of the method has been developed for linear inverse problems adopting an optimization viewpoint. However, many applications require the incorporation of additional constraints on the parameters, e.g. arising due to physical constraints. We propose a new variant of the ensemble Kalman inv…
Ensemble Kalman Inversion as an Inertial Interacting Particle System
Institute for Geometry and Applied Mathematics, RWTH Aachen University（亚琛工业大学几何与应用数学研究所）；Department of Mathematics and Applied Mathematics, University of Pretoria（比勒陀利亚大学数学与应用数学系）；Department of Mathematics, Sapienza University of Rome（罗马第一大学数学系）
Ensemble Kalman inversion (EKI) is a derivative-free optimization method that lies between the deterministic and the probabilistic approaches for inverse problems. EKI iterates the Kalman update of ensemble-based Kalman filters, whose ensemble converges to a minimizer of an objective function. EKI regularizes ill-posed problems by restricting the ensemble to the linear span of the initial ensemble, or by iterating regularization with early stopping. Another regularization approach for EKI, Tikhonov EKI, penalizes the objective function using the $l_2$ penalty term, preventing overfitting in th…
Many data-science problems can be formulated as an inverse problem, where the parameters are estimated by minimizing a proper loss function. When complicated black-box models are involved, derivative-free optimization tools are often needed. The ensemble Kalman filter (EnKF) is a particle-based derivative-free Bayesian algorithm originally designed for data assimilation. Recently, it has been applied to inverse problems for computational efficiency. The resulting algorithm, known as ensemble Kalman inversion (EKI), involves running an ensemble of particles with EnKF update rules so they can co…
The standard probabilistic perspective on machine learning gives rise to empirical risk-minimization tasks that are frequently solved by stochastic gradient descent (SGD) and variants thereof. We present a formulation of these tasks as classical inverse or filtering problems and, furthermore, we propose an efficient, gradient-free algorithm for finding a solution to these problems using ensemble Kalman inversion (EKI). Applications of our approach include offline and online supervised learning with deep neural networks, as well as graph-based semi-supervised learning. The essence of the EKI pr…
The unscented Kalman inversion (UKI) presented in [1] is a general derivative-free approach to solving the inverse problem. UKI is particularly suitable for inverse problems where the forward model is given as a black box and may not be differentiable. The regularization strategy and convergence property of the UKI are thoroughly studied, and the method is demonstrated effectively handling noisy observation data and solving chaotic inverse problems. In this paper, we aim to make the UKI more efficient in terms of computational and memory costs for large scale inverse problems. We take advantag…
The Ensemble Kalman filter (EnKF) was introduced by Evensen in 1994 [10] as a novel method for data assimilation: state estimation for noisily observed time-dependent problems. Since that time it has had enormous impact in many application domains because of its robustness and ease of implementation, and numerical evidence of its accuracy. In this paper we propose the application of an iterative ensemble Kalman method for the solution of a wide class of inverse problems. In this context we show that the estimate of the unknown function that we obtain with the ensemble Kalman method lies in a s…
We propose a new regularisation strategy for the classical ensemble Kalman inversion (EKI) framework. The strategy consists of: (i) an adaptive choice for the regularisation parameter in the update formula in EKI, and (ii) criteria for the early stopping of the scheme. In contrast to existing approaches, our parameter choice does not rely on additional tuning parameters which often have severe effects on the efficiency of EKI. We motivate our approach using the interpretation of EKI as a Gaussian approximation in the Bayesian tempering setting for inverse problems. We show that our parameter c…
The ensemble Kalman inversion (EKI) is a particle based method which has been introduced as the application of the ensemble Kalman filter to inverse problems. In practice it has been widely used as derivative-free optimization method in order to estimate unknown parameters from noisy measurement data. For linear forward models the EKI can be viewed as gradient flow preconditioned by a certain sample covariance matrix. Through the preconditioning the resulting scheme remains in a finite dimensional subspace of the original high-dimensional (or even infinite dimensional) parameter space and can …
The ensemble Kalman inversion (EKI) for the solution of Bayesian inverse problems of type $y = A u +\varepsilon$, with $u$ being an unknown parameter, $y$ a given datum, and $\varepsilon$ measurement noise, is a powerful tool usually derived from a sequential Monte Carlo point of view. It describes the dynamics of an ensemble of particles $\{u^j(t)\}_{j=1}^J$, whose initial empirical measure is sampled from the prior, evolving over an artificial time $t$ towards an approximate solution of the inverse problem, with $t=1$ emulating the posterior, and $t\to\infty$ corresponding to the under-regul…
