High-dimensional mediation analysis (HMA) seeks to uncover complex causal mechanisms involving numerous mediators and plays a crucial role in scientific and social sciences. In this work, we introduce the Generative Adversarial High-dimensional Mediation Network (GAHMN), a novel, scalable structured generative framework designed for causal analysis in high-dimensional settings. GAHMN formulates mediation analysis as dual conditional generative blocks, explicitly capturing mediators' dual roles as outcomes influenced by treatments and as predictors affecting outcomes. Each block integrates a high-dimensional partially linear structure with multi-channel convolutional layers, promoting effective parameter sharing and enhanced representation learning. To induce sparsity and accurate mediator selection, GAHMN employs customized min-max optimization problems with L1 penalties on generator parameters, alongside specially designed optimization algorithms for efficient computation. Unlike existing benchmark methods relying on restrictive parametric assumptions or random-effect specifications, GAHMN flexibly captures heterogeneity, complex distributions, and inter-mediator correlations. With careful design, the computational complexity of GAHMN scales linearly with the number of mediators p, rather than quadratically as in conventional approaches. Theoretical results rigorously ensures estimation consistency, convergence rate, and accurate sparse recovery. GAHMN also serves as a structured generative causal modeling framework, extending to causal decomposition, structural equation modeling, and counterfactual policy evaluation. Extensive experiments confirm GAHMN's superior performance and robustness in synthetic and real-world scenarios.

Online learning methods are designed to establish timely predictive models for machine learning problems. The methods for online learning of nonlinear systems are usually developed in the reproducing kernel Hilbert space (RKHS) associated with Gaussian kernel in which the kernel bandwidth is manually selected and remains steady during the entire modeling process in most cases. This setting may make the learning model rigid and inappropriate for complex data streams. Since the bandwidth appears in a nonlinear term of the kernel model, it raises substantial challenges in the development of learning methods with an adaptive bandwidth. In this article, we propose a novel approach to address this important open issue. By a carefully casted linearization scheme, the nonlinear learning problem is reasonably transformed into a state feedback control problem for a series of controllable systems. Then, by employing optimal control techniques, an effective algorithm is developed, and the parameters in the learning model including kernel bandwidth can be efficiently updated in a real-time manner. By taking advantage of the particular structure of the Gaussian kernel model, a theoretical analysis on the convergence and rationality of the proposed method is also provided. Compared with the kernel algorithms with a fixed bandwidth, our novel learning framework can not only achieve adaptive learning results with a better prediction accuracy but also show performance that is more robust with a faster convergence speed. Encouraging numerical results are provided to demonstrate the advantages of our new method.

In this paper, we propose a novel kernel method for the online identification of stochastic nonlinear spatiotemporal dynamical systems using the robust control approach. By the difference method, the stochastic spatiotemporal (SST) systems driven by multiplicative noise are first transformed into a class of multi-input-multi-output-partially linear kernel models (PLKMs) with heterogeneous random terms. With the help of techniques established for reproducing kernel Hilbert space, the online learning problem is reasonably considered as an output feedback control problem for a group of time varying linear dynamical systems. We develop an effective algorithm to address the learning problem of PLKM and SST systems by employing the model predictive control theory. Compared with the existing learning methods, the new one can achieve adaptive, robust, and fast convergent online modeling performance for the spatiotemporal dynamics with multiplicative noise, which greatly facilitates the characterization of physical characteristics of the system. Moreover, this investigation for the first time addresses the learning problems for SST systems with novel robust control techniques, which can provide some novel insights into the design of kernel machine learning methods from the perspective of optimal control theory. Numerical studies for benchmark systems are presented to illustrate the effectiveness and efficiency of our new method.

The identification of nonlinear spatiotemporal dynamical systems given by partial differential equations has attracted a lot of attention in the past decades. Several methods, such as searching principle-based algorithms, partially linear kernel methods, and coupled lattice methods, have been developed to address the identification problems. However, most existing methods have some restrictions on sampling processes in that the sampling intervals should usually be very small and uniformly distributed in spatiotemporal domains. These are actually not applicable for some practical applications. In this paper, to tackle this issue, a novel kernel-based learning algorithm named integral least square regularization regression (ILSRR) is proposed, which can be used to effectively achieve accurate derivative estimation for nonlinear functions in the time domain. With this technique, a discretization method named inverse meshless collocation is then developed to realize the dimensional reduction of the system to be identified. Thereafter, with this novel inverse meshless collocation model, the ILSRR, and a multiple-kernel-based learning algorithm, a multistep identification method is systematically proposed to address the identification problem of spatiotemporal systems with pointwise nonuniform observations. Numerical studies for benchmark systems with necessary discussions are presented to illustrate the effectiveness and the advantages of the proposed method.

The identification of nonlinear spatiotemporal systems is of significance to engineering practice, since it can always provide useful insight into the underlying nonlinear mechanism and physical characteristics under study. In this paper, nonlinear spatiotemporal system models are transformed into a class of multi-input-multi-output (MIMO) partially linear systems (PLSs), and an effective online identification algorithm is therefore proposed by using a pruning error minimization principle and least square support vector machines. It is shown that many benchmark physical and engineering systems can be transformed into MIMO-PLSs which keep some important physical spatiotemporal relationships and are very helpful in the identification and analysis of the underlying system. Compared with several existing methods, the advantages of the proposed method are that it can make full use of some prior structural information about system physical models, can realize online estimation of the system dynamics, and achieve accurate characterization of some important nonlinear physical characteristics of the system. This would provide an important basis for state estimation, control, optimal analysis, and design of nonlinear distributed parameter systems. The proposed algorithm can also be applied to identification problems of stochastic spatiotemporal dynamical systems. Numeral examples and comparisons are given to demonstrate our results.