Identification of nonlinear spatiotemporal systems via partitioned filtering

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 dynamical systems with bifurcations from noisy partial observation

We propose a statistical machine-learning approach to derive low-dimensional models by integrating noisy time-series data from partial observation of high-dimensional systems, aiming to utilize quantitative data on biological phenomena in the cell. In particular, the method estimates a model from data at different values of a bifurcation parameter in order to characterize biological functions as bifurcation types that are insensitive to system details and experimental errors. The method is tested using artificial data generated from two cell-cycle control system models that exhibit different bifurcations and the learned systems are shown to robustly inherit the bifurcation types.

An adaptive wavelet neural network for spatio-temporal system identification

Starting from the basic concept of coupled map lattices, a new family of adaptive wavelet neural networks (AWNN) is introduced for spatio-temporal system identification, by combining an efficient wavelet representation with a coupled map lattice model. A new orthogonal projection pursuit (OPP) method, coupled with a particle swarm optimization (PSO) algorithm, is proposed for augmenting the proposed network. A novel two-stage hybrid training scheme is developed for constructing a parsimonious network model. In the first stage, by applying the orthogonal projection pursuit algorithm, significant wavelet neurons are adaptively and successively recruited into the network, where adjustable parameters of the associated wavelet neurons are optimized using a particle swarm optimizer. The resultant network model, obtained in the first stage, may however be redundant. In the second stage, an orthogonal least squares algorithm is then applied to refine and improve the initially trained network by removing redundant wavelet neurons from the network. The proposed two-stage hybrid training procedure can generally produce a parsimonious network model, where a ranked list of wavelet neurons, according to the capability of each neuron to represent the total variance in the system output signal is produced. Two spatio-temporal system identification examples are presented to demonstrate the performance of the proposed new modelling framework.