Sparse linear estimation of fluid flows using data-driven proper orthogonal decomposition (POD) basis is systematically explored in this work. Fluid flows are manifestations of nonlinear multiscale partial differential equations (PDE) dynamical systems with inherent scale separation that impact the system dimensionality. Given that sparse reconstruction is inherently an ill-posed problem, the most successful approaches require the knowledge of the underlying low-dimensional space spanning the manifold in which the system resides. In this paper, we adopt an approach that learns basis from singular value decomposition (SVD) of training data to recover sparse information. This results in a set of four design parameters for sparse recovery, namely, the choice of basis, system dimension required for sufficiently accurate reconstruction, sensor budget and their placement. The choice of design parameters implicitly determines the choice of algorithm as either l 2 minimization reconstruction or sparsity promoting l 1 minimization reconstruction. In this work, we systematically explore the implications of these design parameters on reconstruction accuracy so that practical recommendations can be identified. We observe that greedy-smart sensor placement, particularly interpolation points from the discrete empirical interpolation method (DEIM), provide the best balance of computational complexity and accurate reconstruction.

Reconstruction of fine-scale information from sparse data is often needed in practical fluid dynamics where the sensors are typically sparse and yet, one may need to learn the underlying flow structures or inform predictions through assimilation into data-driven models. Given that sparse reconstruction is inherently an ill-posed problem, the most successful approaches encode the physics into an underlying sparse basis space that spans the manifold to generate well-posedness. To achieve this, one commonly uses a generic orthogonal Fourier basis or a data specific proper orthogonal decomposition (POD) basis to reconstruct from sparse sensor information at chosen locations. Such a reconstruction problem is well-posed as long as the sensor locations are incoherent and can sample the key physical mechanisms. The resulting inverse problem is easily solved using l 2 minimization or if necessary, sparsity promoting l 1 minimization. Given the proliferation of machine learning and the need for robust reconstruction frameworks in the face of dynamically evolving flows, we explore in this study the suitability of non-orthogonal basis obtained from extreme learning machine (ELM) auto-encoders for sparse reconstruction. In particular, we assess the interplay between sensor quantity and sensor placement in a given system dimension for accurate reconstruction of canonical fluid flows in comparison to POD-based reconstruction.

The reconstruction of fine-scale information from sparse data measured at irregular locations is often needed in many diverse applications, including numerous instances of practical fluid dynamics observed in natural environments. This need is driven by tasks such as data assimilation or the recovery of fine-scale knowledge including models from limited data. Sparse reconstruction is inherently badly represented when formulated as a linear estimation problem. Therefore, the most successful linear estimation approaches are better represented by recovering the full state on an encoded low-dimensional basis that effectively spans the data. Commonly used low-dimensional spaces include those characterized by orthogonal Fourier and data-driven proper orthogonal decomposition (POD) modes. This article deals with the use of linear estimation methods when one encounters a non-orthogonal basis. As a representative thought example, we focus on linear estimation using a basis from shallow extreme learning machine (ELM) autoencoder networks that are easy to learn but non-orthogonal and which certainly do not parsimoniously represent the data, thus requiring numerous sensors for effective reconstruction. In this paper, we present an efficient and robust framework for sparse data-driven sensor placement and the consequent recovery of the higher-resolution field of basis vectors. The performance improvements are illustrated through examples of fluid flows with varying complexity and benchmarked against well-known POD-based sparse recovery methods.

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