Efficient approximation of solutions of parametric linear transport equations by ReLU DNNs

We demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For non-smooth initial conditions, the solutions of these PDEs are high-dimensional and non-smooth. Therefore, approximation of these functions suffers from a curse of dimension. We demonstrate that through their inherent compositionality deep neural networks can resolve the characteristic flow underlying the transport equations and thereby allow approximation rates independent of the parameter dimension.

An actor-critic-based portfolio investment method inspired by benefit-risk optimization

How to get maximal benefit within a range of risk in securities market is a very interesting and widely concerned issue. Meanwhile, as there are many complex factors that affect securities’ activity, such as the risk and uncertainty of the benefit, it is very difficult to establish an appropriate model for investment. Aiming at solving the curse of dimension and model disaster caused by the problem, we use the approximate dynamic programming to set up a Markov decision model for the multi-time segment portfolio with transaction cost. A model-based actor-critic algorithm under uncertain environment is proposed, where the optimal value function is obtained by iteration on the basis of the constrained risk range and a limited number of funds, and the optimal investment of each period is solved by using the dynamic planning of limited number of fund ratio. The experiment indicated that the algorithm could get a stable investment, and the income could grow steadily.

Beating the curse of dimension with accurate statistics for the Fokker–Planck equation in complex turbulent systems

Solving the Fokker–Planck equation for high-dimensional complex dynamical systems is an important issue. Recently, the authors developed efficient statistically accurate algorithms for solving the Fokker–Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures, which contain many strong non-Gaussian features such as intermittency and fat-tailed probability density functions (PDFs). The algorithms involve a hybrid strategy with a small number of samples L, where a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious Gaussian kernel density estimation in the remaining low-dimensional subspace. In this article, two effective strategies are developed and incorporated into these algorithms. The first strategy involves a judicious block decomposition of the conditional covariance matrix such that the evolutions of different blocks have no interactions, which allows an extremely efficient parallel computation due to the small size of each individual block. The second strategy exploits statistical symmetry for a further reduction of L. The resulting algorithms can efficiently solve the Fokker–Planck equation with strongly non-Gaussian PDFs in much higher dimensions even with orders in the millions and thus beat the curse of dimension. The algorithms are applied to a 1,000-dimensional stochastic coupled FitzHugh–Nagumo model for excitable media. An accurate recovery of both the transient and equilibrium non-Gaussian PDFs requires only L=1 samples! In addition, the block decomposition facilitates the algorithms to efficiently capture the distinct non-Gaussian features at different locations in a 240-dimensional two-layer inhomogeneous Lorenz 96 model, using only L=500 samples.

A Prediction Model Based on ISOMAP for Software Defects

To improve and guarantee the quality of software, it is very necessary to effectively predicting modules with defects in the software. There are usually more measure attributes in software quality prediction, which often leads to the curse of dimension. To do this, a new algorithm based on ISOMAP was presented to predict software defect, which combined manifold learning algorithms and classification methods. In the model, the high dimensional software metrics attribute data were firstly mapped into the low dimensional space through ISOMAP. Then the low dimensional features were classified with KNN, SVM and NB. Experiments demonstrate that the new model progresses the prediction precision of software defects as well as great improves the efficiency of the algorithm.