We investigate the problem of learning an evolution equation directly from some given data. This work develops a learning algorithm to identify the terms in the underlying partial differential equations and to approximate the coefficients of the terms only using data. The algorithm uses sparse optimization in order to perform feature selection and parameter estimation. The features are data driven in the sense that they are constructed using nonlinear algebraic equations on the spatial derivatives of the data. Several numerical experiments show the proposed method's robustness to data noise and size, its ability to capture the true features of the data, and its capability of performing additional analytics. Examples include shock equations, pattern formation, fluid flow and turbulence, and oscillatory convection.

We investigate the approximate dynamics of several differential equations when the solutions are restricted to a sparse subset of a given basis. The restriction is enforced at every time step by simply applying soft thresholding to the coefficients of the basis approximation. By reducing or compressing the information needed to represent the solution at every step, only the essential dynamics are represented. In many cases, there are natural bases derived from the differential equations, which promote sparsity. We find that our method successfully reduces the dynamics of convection equations, diffusion equations, weak shocks, and vorticity equations with high-frequency source terms.multiphysics | multiscale | optimization I n this work, we investigate the approximate dynamics of various partial differential equations (PDEs) whose solutions exhibit behaviors on multiple spatial scales. These scales may interact with one another in a nonlinear manner as they evolve. Many physical equations contain multiscale (as well as multiphysics) phenomena, such as the homogenization problems from material science and chemistry and multiscale systems in biology, computational electrodynamics, fluid dynamics, and atmospheric and oceanic sciences. In some cases, the physical laws used in the model can range from molecular dynamics on the fine scale to classical mechanics on the large scale. In other cases, the equations themselves contain high-wavenumber oscillations that separate into discrete scales, on top of the smooth underlying behavior of the system.The main source of difficulty in multiscale computation is that accurate simulation of the system requires all phenomena to be fully resolved. The smaller spatial scales influence the global solutions; thus, they cannot be ignored in the numerical computation. In some cases, it is possible to derive an analytical equation for the effect of small scales on the solution (1, 2). In practice, however, it may not be possible to derive a simple expression that represents the fine-scale behavior. Many problemdependent methods have been proposed in the literature, whereas a few provide a general methodology for modeling the macroscopic and microscopic processes that yield multiscale models. For example, some general methods include the heterogeneous multiscale method (3), the equation-free method (4), multiscale methods for elliptical problems (5), multiscale finite element methods (6, 7), and the sparse transform method (8). An overview of general multiscale approaches is provided in ref. 9. A key difference between our method and other methods (3-5) is that we are directly resolving all the significant scales in the solution. By contrast, the other methods (3-5) directly resolve only the coarse scales of the solution, and they separately "reconstruct" the fine-scale solution (as well as its effect on the coarse scales).From the perspective of mathematics, multiscale methods began with representation of a function using a global basis, such as a Taylor series or Fourier series. More ...

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One way to understand time-series data is to identify the underlying dynamical system which generates it. This task can be done by selecting an appropriate model and a set of parameters which best fits the dynamics while providing the simplest representation (i.e. the smallest amount of terms). One such approach is the sparse identification of nonlinear dynamics framework [6] which uses a sparsity-promoting algorithm that iterates between a partial least-squares fit and a thresholding (sparsity-promoting) step. In this work, we provide some theoretical results on the behavior and convergence of the algorithm proposed in [6]. In particular, we prove that the algorithm approximates local minimizers of an unconstrained 0 -penalized leastsquares problem. From this, we provide sufficient conditions for general convergence, rate of convergence, and conditions for one-step recovery. Examples illustrate that the rates of convergence are sharp. In addition, our results extend to other algorithms related to the algorithm in [6], and provide theoretical verification to several observed phenomena.