Time Series Forecasting Using Manifold Learning

We address a three-tier numerical framework based on manifold learning for the forecasting of high-dimensional time series. At the first step, we embed the time series into a reduced low-dimensional space using a nonlinear manifold learning algorithm such as Locally Linear Embedding and Diffusion Maps. At the second step, we construct reduced-order regression models on the manifold, in particular Multivariate Autoregressive (MVAR) and Gaussian Process Regression (GPR) models, to forecast the embedded dynamics. At the final step, we lift the embedded time series back to the original high-dimensional space using Radial Basis Functions interpolation and Geometric Harmonics. For our illustrations, we test the forecasting performance of the proposed numerical scheme with four sets of time series: three synthetic stochastic ones resembling EEG signals produced from linear and nonlinear stochastic models with different model orders, and one real-world data set containing daily time series of 10 key foreign exchange rates (FOREX) spanning the time period 03/09/2001-29/10/2020. The forecasting performance of the proposed numerical scheme is assessed using the combinations of manifold learning, modelling and lifting approaches. We also provide a comparison with the Principal Component Analysis algorithm as well as with the naive random walk model and the MVAR and GPR models trained and implemented directly in the high-dimensional space.

Numerical Solution of Stiff Ordinary Differential Equations with Random Projection Neural Networks

We propose a numerical scheme based on Random Projection Neural Networks (RPNN) for the solution of Ordinary Differential Equations (ODEs) with a focus on stiff problems. In particular, we use an Extreme Learning Machine, a single-hidden layer Feedforward Neural Network with Radial Basis Functions which widths are uniformly distributed random variables, while the values of the weights between the input and the hidden layer are set equal to one. The numerical solution is obtained by constructing a system of nonlinear algebraic equations, which is solved with respect to the output weights using the Gauss-Newton method. For our illustrations, we apply the proposed machine learning approach to solve two benchmark stiff problems, namely the Rober and the van der Pol ones (the latter with large values of the stiffness parameter), and we perform a comparison with well-established methods such as the adaptive Runge-Kutta method based on the Dormand-Prince pair, and a variable-step variable-order multistep solver based on numerical differentiation formulas, as implemented in the \texttt{ode45} and \texttt{ode15s} MATLAB functions, respectively. We show that our proposed scheme yields good numerical approximation accuracy without being affected by the stiffness, thus outperforming in same cases the \texttt{ode45} and \texttt{ode15s} functions. Importantly, upon training using a fixed number of collocation points, the proposed scheme approximates the solution in the whole domain in contrast to the classical time integration methods.