49 research outputs found
Empirical Risk Minimization over Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Linear Kolmogorov Partial Differential Equations with Unbounded Initial Functions
Deep learning algorithms have been successfully applied to numerically solve
linear Kolmogorov partial differential equations (PDEs). A recent research
shows that the empirical risk minimization~(ERM) over deep artificial neural
networks overcomes the curse of dimensionality in the numerical approximation
of linear Kolmogorov PDEs with bounded initial functions. However, the initial
functions may be unbounded in many applications such as the Black Scholes PDEs
in pricing call options. In this paper, we extend this result to the cases
involving unbounded initial functions. We prove that for -dimensional linear
Kolmogorov PDEs with unbounded initial functions, under suitable assumptions,
the number of training data and the size of the artificial neural network
required to achieve an accuracy for the ERM grow polynomially in
both and . Moreover, we verify that the required
assumptions hold for Black-Scholes PDEs and heat equations which are two
important cases of linear Kolmogorov PDEs
Competition Between Homophily and Information Entropy Maximization in Social Networks
In social networks, it is conventionally thought that two individuals with
more overlapped friends tend to establish a new friendship, which could be
stated as homophily breeding new connections. While the recent hypothesis of
maximum information entropy is presented as the possible origin of effective
navigation in small-world networks. We find there exists a competition between
information entropy maximization and homophily in local structure through both
theoretical and experimental analysis. This competition means that a newly
built relationship between two individuals with more common friends would lead
to less information entropy gain for them. We conjecture that in the evolution
of the social network, both of the two assumptions coexist. The rule of maximum
information entropy produces weak ties in the network, while the law of
homophily makes the network highly clustered locally and the individuals would
obtain strong and trust ties. Our findings shed light on the social network
modeling from a new perspective
Analysis of the Generalization Error of deep learning based on Randomized Quasi-Monte Carlo for Solving Linear Kolmogorov PDEs
Deep learning algorithms have been widely used to solve linear Kolmogorov
partial differential equations~(PDEs) in high dimensions, where the loss
function is defined as a mathematical expectation. We propose to use the
randomized quasi-Monte Carlo (RQMC) method instead of the Monte Carlo (MC)
method for computing the loss function. In theory, we decompose the error from
empirical risk minimization~(ERM) into the generalization error and the
approximation error. Notably, the approximation error is independent of the
sampling methods. We prove that the convergence order of the mean
generalization error for the RQMC method is for
arbitrarily small , while for the MC method it is
for arbitrarily small . Consequently, we
find that the overall error for the RQMC method is asymptotically smaller than
that for the MC method as increases. Our numerical experiments show that
the algorithm based on the RQMC method consistently achieves smaller relative
error than that based on the MC method
A Hesitant Fuzzy Linguistic Multicriteria Decision-Making Method with Interactive Criteria and Its Application to Renewable Energy Projects Selection
A variety of multicriteria decision-making (MCDM) methods for renewable energy projects evaluation have been proposed, of which the premise of using these methods is to assume that the criteria are independent of each other. However, it may be difficult or costly to build independent criteria set in some cases because renewable energy planning is to pursue a balance of economic, social, and environmental goals, which makes the existence of interaction among criteria be of great possibility. In this paper, we consider a highly ambiguous decision situation, where the experts are allowed to give the evaluations in the form of hesitant fuzzy linguistic terms set (HFLTS). We build a hesitant fuzzy linguistic decision-making model handling the interaction among criteria from the perspective of distance measure and apply it to renewable energy projects selection. The proposed method can consider more fuzzy factors and deal with the interaction among criteria more approximately. It can reduce the decision pressure and improve the decision-making efficiency because the decision makers are allowed to express their preference in form of HFLTS and a decision criteria set of which the criteria are independent of each other is not necessary
Parametric study on the water impacting of a free-falling symmetric wedge based on the extended von Karman's momentum theory
The present study is concerned with the peak acceleration azmax occurring
during the water impact of a symmetric wedge. This aspect can be important for
design considerations of safe marine vehicles. The water-entry problem is
firstly studied numerically using the finite-volume discretization of the
incompressible Navier-Stokes equations and the volume-of-fluid method to
capture the air-water interface. The choice of the mesh size and time-step is
validated by comparison with experimental data of a free fall water-entry of a
wedge. The key original contribution of the article concerns the derivation of
a relationship for azmax (as well as the correlated parameters when azmax
occurs), the initial velocity, the deadrise angle and the mass of the wedge
based on the transformation of von Karman momentum theory which is extended
with the inclusion of the pile-up effect. The pile-up coefficient, which has
been proven dependent on the deadrise angle in the case of water-entry with a
constant velocity, is then investigated for the free fall motion and the
dependence law derived from Dobrovol'skaya is still valid for varying deadrise
angle. Reasonable good theoretical estimates of the kinematic parameters are
provided for a relatively wide range of initial velocity, deadrise angle and
mass using the extended von Karman momentum theory which is the combination of
the original von Karman method and Dobrovol'skaya's solution and this
theoretical approach can be extended to predict the kinematic parameters during
the whole impacting phase.Comment: arXiv admin note: text overlap with arXiv:2207.1041
Effects of wave parameters on load reduction performance for amphibious aircraft with V-hydrofoil
An investigation of the influence of the hydrofoil on load reduction
performance during an amphibious aircraft landing on still and wavy water is
conducted by solving the Unsteady Reynolds-Averaged Navier-Stokes equations
coupled with the standard turbulence model in this paper. During the
simulations, the numerical wave tank is realized by using the velocity-inlet
boundary wave maker coupled with damping wave elimination technique on the
outlet, while the volume of fluid model is employed to track the water-air
interface. Subsequently, the effects of geometric parameters of hydrofoil have
been first discussed on still water, which indicates the primary factor
influencing the load reduction is the static load coefficient of hydrofoil.
Furthermore, the effects of descent velocity, wave length and wave height on
load reduction are comprehensively investigated. The results show that the
vertical load reduces more than 55 at the early stage of landing on the
still water through assembling the hydrofoil for different descent velocity
cases. Meanwhile, for the amphibious aircraft with high forward velocity, the
bottom of the fuselage will come into close contact with the first wave when
landing on crest position, and then the forebody will impact the next wave
surface with extreme force. In this circumstance, the load reduction rate
decreases to around 30, which will entail a further decline with the
increase of wave length or wave height