Convergence of weak-SINDy Surrogate Models

In this paper, we give an in-depth error analysis for surrogate models generated by a variant of the Sparse Identification of Nonlinear Dynamics (SINDy) method. We start with an overview of a variety of non-linear system identification techniques, namely, SINDy, weak-SINDy, and the occupation kernel method. Under the assumption that the dynamics are a finite linear combination of a set of basis functions, these methods establish a matrix equation to recover coefficients. We illuminate the structural similarities between these techniques and establish a projection property for the weak-SINDy technique. Following the overview, we analyze the error of surrogate models generated by a simplified version of weak-SINDy. In particular, under the assumption of boundedness of a composition operator given by the solution, we show that (i) the surrogate dynamics converges towards the true dynamics and (ii) the solution of the surrogate model is reasonably close to the true solution. Finally, as an application, we discuss the use of a combination of weak-SINDy surrogate modeling and proper orthogonal decomposition (POD) to build a surrogate model for partial differential equations (PDEs)

Anderson acceleration with approximate calculations: applications to scientific computing

We provide rigorous theoretical bounds for Anderson acceleration (AA) that allow for efficient approximate calculations of the residual, which reduce computational time and memory storage while maintaining convergence. Specifically, we propose a reduced variant of AA, which consists in projecting the least squares to compute the Anderson mixing onto a subspace of reduced dimension. The dimensionality of this subspace adapts dynamically at each iteration as prescribed by computable heuristic quantities guided by the theoretical error bounds. The use of the heuristic to monitor the error introduced by approximate calculations, combined with the check on monotonicity of the convergence, ensures the convergence of the numerical scheme within a prescribed tolerance threshold on the residual. We numerically assess the performance of AA with approximate calculations on: (i) linear deterministic fixed-point iterations arising from the Richardson's scheme to solve linear systems with open-source benchmark matrices with various preconditioners and (ii) non-linear deterministic fixed-point iterations arising from non-linear time-dependent Boltzmann equations.Comment: 23 pages, 3 figures, 1 tabl

Streaming Compression of Scientific Data via weak-SINDy

In this paper a streaming weak-SINDy algorithm is developed specifically for compressing streaming scientific data. The production of scientific data, either via simulation or experiments, is undergoing an stage of exponential growth, which makes data compression important and often necessary for storing and utilizing large scientific data sets. As opposed to classical ``offline" compression algorithms that perform compression on a readily available data set, streaming compression algorithms compress data ``online" while the data generated from simulation or experiments is still flowing through the system. This feature makes streaming compression algorithms well-suited for scientific data compression, where storing the full data set offline is often infeasible. This work proposes a new streaming compression algorithm, streaming weak-SINDy, which takes advantage of the underlying data characteristics during compression. The streaming weak-SINDy algorithm constructs feature matrices and target vectors in the online stage via a streaming integration method in a memory efficient manner. The feature matrices and target vectors are then used in the offline stage to build a model through a regression process that aims to recover equations that govern the evolution of the data. For compressing high-dimensional streaming data, we adopt a streaming proper orthogonal decomposition (POD) process to reduce the data dimension and then use the streaming weak-SINDy algorithm to compress the temporal data of the POD expansion. We propose modifications to the streaming weak-SINDy algorithm to accommodate the dynamically updated POD basis. By combining the built model from the streaming weak-SINDy algorithm and a small amount of data samples, the full data flow could be reconstructed accurately at a low memory cost, as shown in the numerical tests