Multiscale and nonlocal equations have demonstrated their ability to model accurately the dynamics of complex systems (e.g., the Boltzmann equation). Their direct numerical simulation is prohibitively expensive and a reduced continuum model can significantly reduce the difficulty. However, for problems without a clear scale separation, model reduction of the system becomes non-trivial. We have developed symbolic machine learning to achieve the model recovery of multiscale and nonlocal equations to quantify the scale separation for ease of model reduction [pdf]. Extreme events and related anomalous statistics are fascinating phenomena observed in a wide class of complex nonlinear systems. The main difficulties for accurate quantification and prediction of such extreme events include the lack of a complete understanding of the physical process and the expensive numerical simulation. Practically, the available data for training is often limited with partial observation in a short period. We have developed a deep learning technique to recover missing dynamics in [pdf]. Leveraging the theories of deep learning for regression, we aim at a systematic error analysis of data-driven modeling problems based on deep learning. The first step of convergence analysis was proposed in [pdf], from the viewpoint of approximation theory and time-discretization convergence.

In the future, we will apply symbolic machine learning in climate and ocean studies. The climate system of our Earth is extremely complex including coupling effects from the atmosphere, ocean, and land together with human activities. The main challenges in accurate quantification, prediction, and state estimation of these phenomena include the lack of a complete understanding of the physical process and the expensive computational cost. We aim to address these challenges by exploiting AI to discover the unresolved dynamics from historical data and develop numerical strategies to make the fast computation of high-dimensional equations available. Unlike existing black-box machine learning, we will design AI algorithms to recover the explicit expressions of dynamical equations and related nonlinear response operators, enabling mathematical analysis of tipping points. we will illustrate the crucial strategies for the analysis and prediction of tipping-point climate change with these explicit estimations using real data sets for validation, e.g., El Nino-Southern Oscillation and Madden-Julian Oscillation. Similarly, we will also apply symbolic machine learning to model biological reactions at the system-level based on experimental measurements. Graphical visualization of these projects is given in the figure above.