39 research outputs found
Optimal probabilities and controls for reflecting diffusion processes
A solution to the optimal problem for determining vector fields which
maximize (resp. minimize) the transition probabilities from one location to
another for a class of reflecting diffusion processes is obtained in the
present paper. The approach is based on a representation for the transition
probability density functions. The optimal transition probabilities under the
constraint that the drift vector field is bounded by a constant are studied in
terms of the HJB equation. In dimension one, the optimal reflecting diffusion
processes and the bang-bang diffusion processes are considered. We demonstrate
by simulations that, even in this special case, the optimal diffusion processes
exhibit an interesting feature of phase transitions. We also solve an optimal
stochastic control problem for a class of stochastic control problems involving
diffusion processes with reflection.Comment: 20 Pages, 2 figure
Deep Generative Modeling with Backward Stochastic Differential Equations
This paper proposes a novel deep generative model, called BSDE-Gen, which
combines the flexibility of backward stochastic differential equations (BSDEs)
with the power of deep neural networks for generating high-dimensional complex
target data, particularly in the field of image generation. The incorporation
of stochasticity and uncertainty in the generative modeling process makes
BSDE-Gen an effective and natural approach for generating high-dimensional
data. The paper provides a theoretical framework for BSDE-Gen, describes its
model architecture, presents the maximum mean discrepancy (MMD) loss function
used for training, and reports experimental results.Comment: 17 pages, 5 figure
It Ain't That Bad: Understanding the Mysterious Performance Drop in OOD Generalization for Generative Transformer Models
Generative Transformer-based models have achieved remarkable proficiency on
solving diverse problems. However, their generalization ability is not fully
understood and not always satisfying. Researchers take basic mathematical tasks
like n-digit addition or multiplication as important perspectives for
investigating their generalization behaviors. Curiously, it is observed that
when training on n-digit operations (e.g., additions) in which both input
operands are n-digit in length, models generalize successfully on unseen
n-digit inputs (in-distribution (ID) generalization), but fail miserably and
mysteriously on longer, unseen cases (out-of-distribution (OOD)
generalization). Studies try to bridge this gap with workarounds such as
modifying position embedding, fine-tuning, and priming with more extensive or
instructive data. However, without addressing the essential mechanism, there is
hardly any guarantee regarding the robustness of these solutions. We bring this
unexplained performance drop into attention and ask whether it is purely from
random errors. Here we turn to the mechanistic line of research which has
notable successes in model interpretability. We discover that the strong ID
generalization stems from structured representations, while behind the
unsatisfying OOD performance, the models still exhibit clear learned algebraic
structures. Specifically, these models map unseen OOD inputs to outputs with
equivalence relations in the ID domain. These highlight the potential of the
models to carry useful information for improved generalization
Explicit solutions for a class of nonlinear backward stochastic differential equations and their nodal sets
In this paper, we investigate a class of nonlinear backward stochastic
differential equations (BSDEs) arising from financial economics, and give
specific information about the nodal sets of the related solutions. As
applications, we are able to obtain the explicit solutions to an interesting
class of nonlinear BSDEs including the k-ignorance BSDE arising from the
modeling of ambiguity of asset pricing
Large Language Models at Work in China's Labor Market
This paper explores the potential impacts of large language models (LLMs) on
the Chinese labor market. We analyze occupational exposure to LLM capabilities
by incorporating human expertise and LLM classifications, following Eloundou et
al. (2023)'s methodology. We then aggregate occupation exposure to the industry
level to obtain industry exposure scores. The results indicate a positive
correlation between occupation exposure and wage levels/experience premiums,
suggesting higher-paying and experience-intensive jobs may face greater
displacement risks from LLM-powered software. The industry exposure scores
align with expert assessments and economic intuitions. We also develop an
economic growth model incorporating industry exposure to quantify the
productivity-employment trade-off from AI adoption. Overall, this study
provides an analytical basis for understanding the labor market impacts of
increasingly capable AI systems in China. Key innovations include the
occupation-level exposure analysis, industry aggregation approach, and economic
modeling incorporating AI adoption and labor market effects. The findings will
inform policymakers and businesses on strategies for maximizing the benefits of
AI while mitigating adverse disruption risks
Enhance Diamond Coating Adhesion by Oriented Interlayer Microcracking
In this paper, we report a microcrack toughening mechanism for enhancing the adhesion of diamondcoating. The oriented microcracks were formed within the TiC interlayer to dissipate strain energy and accommodate deformation via the crack opening-closing mechanism, thus enhancing the coating/substrate interfacial toughness. The delamination of diamondcoating was effectively prevented when the parallel microcracks were confined within the interlayer and arrested at interfaces of coating/interlayer/substrate. Density functional theory calculations revealed that the highly anisotropicfracture strength of the TiC phase energetically favors crack initiation and propagation along (100) planes only, which are 54.7° away from the interface. These microcracks are constrained inside the interlayer by the two strong interfaces in the substrate/interlayer/coating system. The new microcrack toughening mechanism with these combined features has a wide application to enhance the adhesion of thin-film coatings