Time-series forecasting using manifold learning, radial basis function interpolation, and geometric harmonics.

Area of Science:

• Computational Mathematics
• Data Science
• Machine Learning

Background:

• High-dimensional time series forecasting and partial differential equation (PDE) modeling face challenges due to the curse of dimensionality.
• Existing surrogate and machine learning models require extensive training data and computational resources.
• Developing efficient reduced-order models is crucial for handling complex dynamical systems.

Purpose of the Study:

• To present a novel three-tier numerical framework for forecasting high-dimensional time series.
• To apply this framework as a reduced-order modeling procedure for solving transient PDE dynamics.
• To mitigate the "curse of dimensionality" in the training of surrogate/machine learning models.

Main Methods:

• Nonlinear manifold learning (local linear embedding, parsimonious diffusion maps) to embed data into a low-dimensional space.
• Construction of reduced-order surrogate models (multivariate autoregressive, Gaussian process regression) on the manifold.
• Radial basis function interpolation and geometric harmonics to solve the pre-image problem and reconstruct high-dimensional data.

Main Results:

• The framework successfully forecasts synthetic time series from stochastic models resembling electroencephalography signals.
• Accurate prediction and propagation of solution profiles for linear parabolic and nonlinear Brusselator PDEs were achieved.
• Effective forecasting of a real-world dataset of daily foreign exchange rates was demonstrated.

Conclusions:

• The proposed data-driven scheme effectively reduces dimensionality for time series forecasting and PDE propagation.
• This approach offers a computationally efficient alternative to traditional high-dimensional modeling techniques.
• The framework shows broad applicability across synthetic, simulated, and real-world complex dynamical systems.