Nearly all model-reduction techniques project the governing equations onto a linear subspace of the original state space. Such subspaces are typically computed using methods such as balanced truncation, rational interpolation, the reduced-basis method, and (balanced) proper orthogonal decomposition (POD). Unfortunately, restricting the state to evolve in a linear subspace imposes a fundamental limitation to the accuracy of the resulting reduced-order model (ROM). In particular, linear-subspace ROMs can be expected to produce low-dimensional models with high accuracy only if the problem admits a fast decaying Kolmogorov n-width (e.g., diffusion-dominated problems). Unfortunately, many problems of interest exhibit a slowly decaying Kolmogorov n-width (e.g., advection-dominated problems). To address this, we propose a novel framework for projecting dynamical systems onto nonlinear manifolds using minimum-residual formulations at the time-continuous and time-discrete levels; the former leads to manifold Galerkin projection, while the latter leads to manifold least-squares Petrov-Galerkin (LSPG) projection. We perform analyses that provide insight into the relationship between these proposed approaches and classical linear-subspace reduced-order models; we also derive a posteriori discrete-time error bounds for the proposed approaches. In addition, we propose a computationally practical approach for computing the nonlinear manifold, which is based on convolutional autoencoders from deep learning. Finally, we demonstrate the ability of the method to significantly outperform even the optimal linear-subspace ROM on benchmark advection-dominated problems, thereby demonstrating the method's ability to overcome the intrinsic n-width limitations of linear subspaces.

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In this study, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the Kronecker product structure of the linear systems. The proposed algorithm efficiently approximates the solutions in low-rank tensor format. Using standard Krylov subspace methods for the data in tensor format is computationally prohibitive due to the rapid growth of tensor ranks during the iterations. To keep tensor ranks low over the entire iteration process, we devise a rank-reduction scheme that can be combined with the iterative algorithm. The proposed rank-reduction scheme identifies an important subspace in the stochastic domain and compresses tensors of high rank on-the-fly during the iterations. The proposed reduction scheme is a multilevel method in that the important subspace can be identified inexpensively in a coarse spatial grid setting. The efficiency of the proposed method is illustrated by numerical experiments on benchmark problems.

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Thickness-dependent bandgap and carrier mobility of two-dimensional (2D) van der Waals (vdW) layered materials make them a promising material as a phototransistor that detects light signals and converts them to electrical signals. Thus far, to achieve a high photoresponsivity of 2D materials, enormous efforts have been made via material and dielectric engineering, as well as modifying device structure. Nevertheless, understanding the effect of interplay between the thickness and the carrier mobility to photoresponsivity is little known. Here, we demonstrate the tunable photoresponsivity (R) of 2D multilayer rhenium disulfide (ReS2), which is an attractive candidate for photodetection among 2D vdW materials owing to its layer-independent direct bandgap and decoupled vdW interaction. The gate bias (VG)-dependent photocurrent generation mechanism and R are presented for the channel thickness range of 1.7–27.5 nm. The high similarity between VG-dependent photocurrent and transconductance features in the ReS2 phototransistors clearly implies the importance of the channel thickness and the operating VG bias condition. Finally, the maximum R was found to be 4.1 × 105 A/W at 14.3 nm with the highest carrier mobility of ∼15.7 cm2⋅V−1⋅s−1 among the fabricated devices after excluding the contact resistance effect. This work sheds light on the strategy of how to obtain the highest R in 2D vdW-based phototransistors.