Sparse learning of partial differential equations with structured dictionary matrix.

This paper presents a "structured" learning approach for the identification of continuous partial differential equation (PDE) models with both constant and spatial-varying coefficients. The identification problem of parametric PDEs can be formulated as an ℓ1/ℓ2-mixed optimization problem by explicitly using block structures. Block-sparsity is used to ensure parsimonious representations of parametric spatiotemporal dynamics. An iterative reweighted ℓ1/ℓ2 algorithm is proposed to solve the ℓ1/ℓ2-mixed optimization problem. In particular, the estimated values of varying coefficients are further used as data to identify functional forms of the coefficients. In addition, a new type of structured random dictionary matrix is constructed for the identification of constant-coefficient PDEs by introducing randomness into a bounded system of Legendre orthogonal polynomials. By exploring the restricted isometry properties of the structured random dictionary matrices, we derive a recovery condition that relates the number of samples to the sparsity and the probability of failure in the Lasso scheme. Numerical examples, such as the Schrödinger equation, the Fisher-Kolmogorov-Petrovsky-Piskunov equation, the Burger equation, and the Fisher equation, suggest that the proposed algorithm is fairly effective, especially when using a limited amount of measurements.

Identifying Community-Bridge Network Structures via Bayesian Learning With Mixed Sparsity Mode.

Identifying structures of complex networks based on time series of nodal data is of considerable interest and significance in many fields of science and engineering. This article presents a sparse Bayesian learning (SBL) method for identifying structures of community-bridge networks, where nodes are grouped to form communities connected via bridges. Using the structural information of such networks with unknown nodal dynamics and community formations, network structure identification is tackled similar to sparse signal reconstruction with mixed sparsity mode. The proposed method is theoretically proved to be convergent. Its superiority to mainstream baselines is demonstrated via extensive experiments without the need for manual adjustment of regularization parameters.

Sparse Bayesian Learning for Switching Network Identification.

Learning dynamical networks based on time series of nodal states is of significant interest in systems science, computer science, and control engineering. Despite recent progress in network identification, most research focuses on static structures rather than switching ones. Therefore, this article develops a method for identifying the structures of switching networks by exploring and leveraging both temporal and spatial structural information that characterizes the switching process. The proposed method employs a new sparse Bayesian learning algorithm based on coupled hyperblocks to estimate unknown switching instants. Experimental results on benchmark artificial and real networks are elaborated to demonstrate the effectiveness and superiority of the proposed method.

Inferring upstream regulatory genes of FOXP3 in human regulatory T cells from time-series transcriptomic data.

The discovery of upstream regulatory genes of a gene of interest still remains challenging. Here we applied a scalable computational method to unbiasedly predict candidate regulatory genes of critical transcription factors by searching the whole genome. We illustrated our approach with a case study on the master regulator FOXP3 of human primary regulatory T cells (Tregs). While target genes of FOXP3 have been identified, its upstream regulatory machinery still remains elusive. Our methodology selected five top-ranked candidates that were tested via proof-of-concept experiments. Following knockdown, three out of five candidates showed significant effects on the mRNA expression of FOXP3 across multiple donors. This provides insights into the regulatory mechanisms modulating FOXP3 transcriptional expression in Tregs. Overall, at the genome level this represents a high level of accuracy in predicting upstream regulatory genes of key genes of interest.