Constructing accurate and computationally efficient surrogate models (or emulators) for predicting dynamical system responses is critical in many engineering domains, yet remains challenging due to the strongly nonlinear and high-dimensional mapping from external excitations and system parameters to system responses. This work introduces a novel Function-on-Function Nonlinear AutoRegressive model with eXogenous inputs (F2NARX), which reformulates the conventional NARX model from a function-on-function regression perspective, inspired by the recently proposed $\mathcal{F}$-NARX method. The proposed framework substantially improves predictive efficiency while maintaining high accuracy. By combining principal component analysis with Gaussian process regression, F2NARX further enables probabilistic predictions of dynamical responses via the unscented transform in an autoregressive manner. The effectiveness of the method is demonstrated through case studies of varying complexity. Results show that F2NARX outperforms state-of-the-art NARX model by orders of magnitude in efficiency while achieving higher accuracy in general. Moreover, its probabilistic prediction capabilities facilitate active learning, enabling accurate estimation of first-passage failure probabilities of dynamical systems using only a small number of training time histories.

In this study, we present and validate the predictive capability of the Physics-Informed Neural Networks (PINNs) methodology for solving a variety of engineering and biological dynamical systems governed by ordinary differential equations (ODEs). While traditional numerical methods a re effective for many ODEs, they often struggle to achieve convergence in problems involving high stiffness, shocks, irregular domains, singular perturbations, high dimensions, or boundary discontinuities. Alternatively, PINNs offer a powerful approach for handling challenging numerical scenarios. In this study, classical ODE problems are employed as controlled testbeds to systematically evaluate the accuracy, training efficiency, and generalization capability under controlled conditions of the PINNs framework. Although not a universal solution, PINNs can achieve superior results by embedding physical laws directly into the learning process. We first analyze the existence and uniqueness properties of several benchmark problems and subsequently validate the PINNs methodology on these model systems. Our results demonstrate that for complex problems to converge to correct solutions, the loss function components data loss, initial condition loss, and residual loss must be appropriately balanced through careful weighting. We further establish that systematic tuning of hyperparameters, including network depth, layer width, activation functions, learning rate, optimization algorithms, w eight initialization schemes, and collocation point sampling, plays a crucial role in achieving accurate solutions. Additionally, embedding prior knowledge and imposing hard constraints on the network architecture, without loss the generality of the ODE system, significantly enhances the predictive capability of PINNs.

We present a neural-network emulator for the thermal and chemical evolution in Population III star formation. The emulator accurately reproduces the thermochemical evolution over a wide density range spanning 21 orders of magnitude (10$^{-3}$-10$^{18}$ cm$^{-3}$), tracking six primordial species: H, H$_2$, e$^{-}$, H$^{+}$, H$^{-}$, and H$_2^{+}$. To handle the broad dynamic range, we partition the density range into five subregions and train separate deep operator networks (DeepONets) in each region. When applied to randomly sampled thermochemical states, the emulator achieves relative errors below 10% in over 90% of cases for both temperature and chemical abundances (except for the rare species H$_2^{+}$). The emulator is roughly ten times faster on a CPU and more than 1000 times faster for batched predictions on a GPU, compared with conventional numerical integration. Furthermore, to ensure robust predictions under many iterations, we introduce a novel timescale-based update method, where a short-timestep update of each variable is computed by rescaling the predicted change over a longer timestep equal to its characteristic variation timescale. In one-zone collapse calculations, the results from the timescale-based method agree well with traditional numerical integration even with many iterations at a timestep as short as 10$^{-4}$ of the free-fall time. This proof-of-concept study suggests the potential for neural network-based chemical emulators to accelerate hydrodynamic simulations of star formation.

We consider the problem of subspace clustering: given points that lie on or near the union of many low-dimensional linear subspaces, recover the subspaces. To this end, one first identifies sets of points close to the same subspace and uses the sets to estimate the subspaces. As the geometric structure of the clusters (linear subspaces) forbids proper performance of general distance based approaches such as K -means, many model-specific methods have been proposed. In this paper, we provide new simple and efficient algorithms for this problem. Our statistical analysis shows that the algorithms are guaranteed exact (perfect) clustering performance under certain conditions on the number of points and the affinity between subspaces. These conditions are weaker than those considered in the standard statistical literature. Experimental results on synthetic data generated from the standard unions of subspaces model demonstrate our theory. We also show that our algorithm performs competitively against state-of-the-art algorithms on real-world applications such as motion segmentation and face clustering, with much simpler implementation and lower computational cost.

Sliced Wasserstein (SW) distance suffers from redundant projections due to independent uniform random projecting directions.