Naichang Ke

Pongpisit Thanasutives, Naichang Ke, Yoshinobu Kawahara

We propose KO-PDE-IDENT, a data-driven framework for identifying parsimonious partial differential equations (PDEs) with false discovery rate (FDR) control. PDE discovery from noisy observations is often hindered by extreme multicollinearity among candidate terms, which causes typical sparse-regression methods to select spurious terms. To address this problem, KO-PDE-IDENT initially mines a support set of potential candidate terms via model-X knockoff filters with finite-sample FDR control, then refines and ranks the surviving PDE alternatives. The framework integrates three components. First, knockoff feature statistics are constructed by coupling $\ell_{0}$-constrained adaptive best-subset selection with SHapley Additive exPlanations (SHAP), yielding an effective and computationally efficient difference statistic. Second, a recursive feature elimination (RFE) procedure removes terms whose marginal contributions are dispensable and assesses statistical necessity through knockoff-perturbed hypothesis testing. Third, the final model selection is formulated as a multi-criteria decision-making (MCDM) problem, where the optimal governing equation is the alternative that best balances a wide range of criteria such as predictive accuracy, model complexity and coefficient uncertainty. We validate KO-PDE-IDENT on five canonical PDEs under severe noise corruption. Empirical results show that our framework can exactly recover the true PDE structure, eliminating false discoveries while retaining all true underlying terms, with low coefficient estimation error.

Naichang Ke, Ryogo Tanaka, Yoshinobu Kawahara

We consider an operator-based latent Markov representation of a stochastic nonlinear dynamical system, where the stochastic evolution of the latent state embedded in a reproducing kernel Hilbert space is described with the corresponding transfer operator, and develop a spectral method to learn this representation based on the theory of stochastic realization. The embedding may be learned simultaneously using reproducing kernels, for example, constructed with feed-forward neural networks. We also address the generalization of sequential state-estimation (Kalman filtering) in stochastic nonlinear systems, and of operator-based eigen-mode decomposition of dynamics, for the representation. Several examples with synthetic and real-world data are shown to illustrate the empirical characteristics of our methods, and to investigate the performance of our model in sequential state-estimation and mode decomposition.