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We are concerned with high-dimensional coupled FBSDE systems approximated by the deep BSDE method of Han et al. (2018). It was shown by Han and Long (2020) that the errors induced by the deep BSDE method admit a posteriori estimate depending on the loss function, whenever the backward equation only couples into the forward diffusion through the Y process. We generalize this result to fully-coupled drift coefficients, and give sufficient conditions for convergence under standard assumptions. The resulting conditions are directly verifiable for any equation. Consequently, unlike in earlier theory, our convergence analysis enables the treatment of FBSDEs stemming from stochastic optimal control problems. In particular, we provide a theoretical justification for the non-convergence of the deep BSDE method observed in recent literature, and present direct guidelines for when convergence can be guaranteed in practice. Our theoretical findings are supported by several numerical experiments in high-dimensional settings.

The thermo-hydro-mechanical of unsaturated soils plays a significant role in dynamic shear banding and fracturing. In this article, we propose a thermo-hydro-mechanical material model in the periporomechanics paradigm to model shear banding and crack triggered by temperature. Periporomechanics is a nonlocal framework for the mechanics of unsaturated soil where a length scale dictates the nonlocal interaction between material points. Periporomechanics unites continuous and discontinuous deformation and fluid flow in porous media. As a new contribution, we incorporate the thermo-hydro-mechanical material model in the periporomechanics through the correspondence principle for modeling shear banding and cracking in unsaturated porous media. The stabilized PPM correspondence principle that mitigates the multiphase zero-energy mode instability is augmented. At the global level, we have numerically implemented the periporomechanics paradigm through an explicit Lagrangian meshfree algorithm in the global level. At the local level, we impose the return mapping algorithm to implement the thermo-hydro-mechanical constitutive model. We present numerical examples to demonstrate the efficacy and robustness of proposed periporomechanics for modeling the shear banding bifurcation and crack in unsaturated porous media triggered by temperature.

Unisolvence of unsymmetric Kansa collocation is still a substantially open problem. We prove that Kansa matrices with MultiQuadrics and Inverse MultiQuadrics for the Dirichlet problem of the Poisson equation are almost surely nonsingular, when the collocation points are chosen by any continuous random distribution in the domain interior and arbitrarily on its boundary.

On the Boolean domain, there is a class of symmetric signatures called ``Fibonacci gates'' for which a beautiful P-time combinatorial algorithm has been designed for the corresponding $\operatorname{Holant}$ problems. In this work, we give a combinatorial view for $\operatorname{Holant}(\mathcal{F})$ problems on a domain of size 3 where $\mathcal{F}$ is a set of arity 3 functions with inputs taking values on the domain of size 3 and the functions share some common properties. The combinatorial view can also be extended to the domain of size 4. Specifically, we extend the definition of "Fibonacci gates" to the domain of size 3 and the domain of size 4. Moreover, we give the corresponding combinatorial algorithms.

We introduce a high-order space-time approximation of the Shallow Water Equations with sources that is invariant-domain preserving (IDP) and well-balanced with respect to rest states. The employed time-stepping technique is a novel explicit Runge-Kutta (ERK) approach which is an extension of the class of ERK-IDP methods introduced by Ern and Guermond (SIAM J. Sci. Comput. 44(5), A3366--A3392, 2022) for systems of non-linear conservation equations. The resulting method is then numerically illustrated through verification and validation.

Generative Autoregressive Neural Networks (ARNNs) have recently demonstrated exceptional results in image and language generation tasks, contributing to the growing popularity of generative models in both scientific and commercial applications. This work presents an exact mapping of the Boltzmann distribution of binary pairwise interacting systems into autoregressive form. The resulting ARNN architecture has weights and biases of its first layer corresponding to the Hamiltonian's couplings and external fields, featuring widely used structures such as the residual connections and a recurrent architecture with clear physical meanings. Moreover, its architecture's explicit formulation enables the use of statistical physics techniques to derive new ARNNs for specific systems. As examples, new effective ARNN architectures are derived from two well-known mean-field systems, the Curie-Weiss and Sherrington-Kirkpatrick models, showing superior performance in approximating the Boltzmann distributions of the corresponding physics model compared to other commonly used architectures. The connection established between the physics of the system and the neural network architecture provides a means to derive new architectures for different interacting systems and interpret existing ones from a physical perspective.

The Dvoretzky--Kiefer--Wolfowitz--Massart inequality gives a sub-Gaussian tail bound on the supremum norm distance between the empirical distribution function of a random sample and its population counterpart. We provide a short proof of a result that improves the existing bound in two respects. First, our one-sided bound holds without any restrictions on the failure probability, thereby verifying a conjecture of Birnbaum and McCarty (1958). Second, it is local in the sense that it holds uniformly over sub-intervals of the real line with an error rate that adapts to the behaviour of the population distribution function on the interval.

We propose to improve the convergence properties of the single-reference coupled cluster (CC) method through an augmented Lagrangian formalism. The conventional CC method changes a linear high-dimensional eigenvalue problem with exponential size into a problem of determining the roots of a nonlinear system of equations that has a manageable size. However, current numerical procedures for solving this system of equations to get the lowest eigenvalue suffer from two practical issues: First, solving the CC equations may not converge, and second, when converging, they may converge to other -- potentially unphysical -- states, which are stationary points of the CC energy expression. We show that both issues can be dealt with when a suitably defined energy is minimized in addition to solving the original CC equations. We further propose an augmented Lagrangian method for coupled cluster (alm-CC) to solve the resulting constrained optimization problem. We numerically investigate the proposed augmented Lagrangian formulation showing that the convergence towards the ground state is significantly more stable and that the optimization procedure is less susceptible to local minima. Furthermore, the computational cost of alm-CC is comparable to the conventional CC method.

Edge AI has been recently proposed to facilitate the training and deployment of Deep Neural Network (DNN) models in proximity to the sources of data. To enable the training of large models on resource-constraint edge devices and protect data privacy, parallel split learning is becoming a practical and popular approach. However, current parallel split learning neglects the resource heterogeneity of edge devices, which may lead to the straggler issue. In this paper, we propose EdgeSplit, a novel parallel split learning framework to better accelerate distributed model training on heterogeneous and resource-constraint edge devices. EdgeSplit enhances the efficiency of model training on less powerful edge devices by adaptively segmenting the model into varying depths. Our approach focuses on reducing total training time by formulating and solving a task scheduling problem, which determines the most efficient model partition points and bandwidth allocation for each device. We employ a straightforward yet effective alternating algorithm for this purpose. Comprehensive tests conducted with a range of DNN models and datasets demonstrate that EdgeSplit not only facilitates the training of large models on resource-restricted edge devices but also surpasses existing baselines in performance.

Graph Convolutional Neural Networks (GCNs) possess strong capabilities for processing graph data in non-grid domains. They can capture the topological logical structure and node features in graphs and integrate them into nodes' final representations. GCNs have been extensively studied in various fields, such as recommendation systems, social networks, and protein molecular structures. With the increasing application of graph neural networks, research has focused on improving their performance while compressing their size. In this work, a plug-in module named Graph Knowledge Enhancement and Distillation Module (GKEDM) is proposed. GKEDM can enhance node representations and improve the performance of GCNs by extracting and aggregating graph information via multi-head attention mechanism. Furthermore, GKEDM can serve as an auxiliary transferor for knowledge distillation. With a specially designed attention distillation method, GKEDM can distill the knowledge of large teacher models into high-performance and compact student models. Experiments on multiple datasets demonstrate that GKEDM can significantly improve the performance of various GCNs with minimal overhead. Furthermore, it can efficiently transfer distilled knowledge from large teacher networks to small student networks via attention distillation.