This paper focuses on the curse of dimensionality in numerical solution stationary Fokker–Planck equation for systems with state-independent excitation. A tensor decomposition approach is combined Chebyshev spectral differentiation to drastically reduce number degrees freedom required maintain accuracy as increases. Following enforcement normality condition via a penalty method, discretized system solved using alternating least squares algorithm. Numerical results variety systems, including...

We formulate low Mach number fluctuating hydrodynamic equations appropriate for modeling diffusive mixing in isothermal mixtures of fluids with different density and transport coefficients. These eliminate the fluctuations pressure associated propagation sound waves by replacing equation state a local thermodynamic constraint. demonstrate that model preserves spatio-temporal spectrum slower fluctuations. develop strictly conservative finite-volume spatial discretization both two three...

Physical simulations are essential tools across critical fields such as mechanical and aerospace engineering, chemistry, meteorology, etc. While neural operators, particularly the Fourier Neural Operator (FNO), have shown promise in predicting simulation results with impressive performance efficiency, they face limitations when handling real-world scenarios involving coupled multi-physics outputs. Current operator methods either overlook correlations between multiple physical processes or...

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This paper addresses the curse of dimensionality in numerical solution stationary Fokker-Planck equations. Combined with Chebyshev spectral differentiation, tensor approach significantly reduces degrees freedom approximation essentially exchange for nonlinearity, such that resulting discretized nonlinear system is solved by alternating least squares. Enforcement normality condition via a penalty method avoids need exploration null space operator. The proposed enables drastic reduction...

In this paper the Fokker-Planck equation is solved by meshless particle partition of unity finite element method with automatic boundary condition enforcement. An improved cover generation algorithm presented to automatically minimize interference between nodes and interior in solution domain. Radial basis functions are used at damp a desired small value boundaries, thus eliminating need penalty for This achieves two important goals: (i) improves conditioning discretized reduced order system...

This paper presents a particle approach based on Markov chain Monte Carlo and the method of characteristics to numerically solve stochastic Liouville equation for nonlinear dynamical systems. theory helps accomplish two important objectives: (i) it provides viable systems with high dimensional state space by using compact representation that is equivalent in measure time varying probability density function state, (ii) automatically extracts domain significance uncertainty (i.e. support...

A tensor decomposition approach combined with Chebyshev spectral differentiation is developed to solve the transient Fokker-Planck equation (FPE) in high dimensional cases. This method drastically reduces degrees of freedom required maintain accuracy approximation as dimensionality increases. The solution sought a single CANDECOMP/PARAFAC form for all times by alternating least squares algorithm. accomplished decoupling spatial dimensions well separation temporal domain from domain. As...

The multi-bead-spring (MBS) model for polymeric liquids is commonly employed to represent the coarse-grained molecular configuration characterized by Fokker-Planck Equation (FPE). For chain consisting of several beads, state space involved in resulting transient FPE would become rather high dimensional. And solving this dimensional a challenging task even on supercomputers due curse dimensionality. In paper, tensor decomposition approach combined with Chebyshev spectral differentiation used...

The Response Surface Method (RSM) is not efficient to solve reliability analysis of complex and computationally high-demanding models. Stochastic (SRSM) was proposed recently. This paper adds some mathematics foundation for the SRSM, compares SRSM RSM. two approaches are applied a nonlinear numerical example statically indeterminate beam problem respectively. results show that can be used efficient, accurate estimate reliability.

The present fractional-order plasticity models for granular soil are mainly established under the triaxial compression condition, due to its difficult in analytically solving fractional differentiation of third stress invariant, e.g., Lode\'s angle. To solve this problem, a three dimensional elastoplastic model based on transformed method, which does not rely analytical solution angle, is proposed. A nonassociated plastic flow rule derived by conducting derivative yielding function with...

This paper considers controlled scalar systems relying on a lossy wireless feedback channel. In contrast with the existing literature, focus is not system controller but transmit power that implemented at side for reporting state to controller. Such problem may be of interest, e.g., remote control drones, where communication costs have considered. Determining policy minimizes combination dynamical cost and transmission energy shown non-trivial optimization problem. It turns out recursive...

Uncertainty forecasting of Earth-orbiting targets is an essential but difficult task, because the underlying Fokker-Planck equation (FPE) defined on a relatively high dimensional state-space, and driven by nonlinear perturbed Keplerian dynamics. In addition, enormously large solution domain required, which time-varying state probability density function (pdf) only occupies tiny fraction at any given time. This has caused FPE for orbital uncertainty propagation to remain unsolved problem....

Artificial intelligence (AI) has immense potential in time series prediction, but most explainable tools have limited capabilities providing a systematic understanding of important features over time. These typically rely on evaluating single point, overlook the ordering inputs, and neglect time-sensitive nature applications. factors make it difficult for users, particularly those without domain knowledge, to comprehend AI model decisions obtain meaningful explanations. We propose CGS-Mask,...

Content-based recommendation systems play a crucial role in delivering personalized content to users the digital world. In this work, we introduce EmbSum, novel framework that enables offline pre-computations of and candidate items while capturing interactions within user engagement history. By utilizing pretrained encoder-decoder model poly-attention layers, EmbSum derives User Poly-Embedding (UPE) Content (CPE) calculate relevance scores between items. actively learns long histories by...

The human brain is a complex and highly dynamic system, our current knowledge of its functional mechanism still very limited. Fortunately, with magnetic resonance imaging (fMRI), we can observe blood oxygen level-dependent (BOLD) changes, reflecting neural activity, to infer states dynamics. In this paper, ask the question whether rep-resented by regional fMRI be predicted. Due success self-attention transformer architecture in sequential auto-regression problems (e.g., language modelling or...

We introduce an adaptively weighted Galerkin approach for elliptic problems where diffusion is dominated by strong convection or reaction terms.In such problems, standard approximations can have unacceptable oscillatory behavior near boundaries unless the computational mesh sufficiently fine.Here we show how weighting equations within variational problem increase accuracy and stability of solutions on under-resolved meshes.Rather than relying specialized finite elements meshes, idea here...

Machine learning models are increasingly used in time series prediction with promising results. The model explanation of falls behind the development and makes less sense to users understanding decisions. This paper proposes ES-Mask, a post-hoc model-agnostic evolutionary strip mask-based saliency approach for applications. ES-Mask designs mask consisting strips same salient value consecutive steps produce binary sustained feature importance scores over easy interpretation series. uses an...