Neural Networks and Stochastic Differential Equations

Influenced by the seminal work, “Physics Informed Neural Networks” by Raissi et al., 2017, there has been a growing interest in solving and parameter estimation of Nonlinear Partial Differential Equations (PDE) with Deep Neural networks in recent years. In fact, this has broadened the pathways and shed light on deep learning of stochastic differential equations (SDE) and stochastic PDE’s (SPDE).In this work, we intend to investigate the current approaches of solving and parameter estimation of the SDE/SPDE with deep neural networks and the possibility of extending them to obtain more accurate/stable solutions with residual systems and/or generative adversarial neural networks. We will also apply these methods to study real-world applications of SDE/SPDE problems. The combination of methods can improve speeds, accuracy, and lessen data-related difficulties in solving Stochastic PDEs. Such improvements can assist a wide array of the sciences in computational research and data sciences

Tasks Makyth Models: Machine Learning Assisted Surrogates for Tipping Points

We present a machine learning (ML)-assisted framework bridging manifold learning, neural networks, Gaussian processes, and Equation-Free multiscale modeling, for (a) detecting tipping points in the emergent behavior of complex systems, and (b) characterizing probabilities of rare events (here, catastrophic shifts) near them. Our illustrative example is an event-driven, stochastic agent-based model (ABM) describing the mimetic behavior of traders in a simple financial market. Given high-dimensional spatiotemporal data -- generated by the stochastic ABM -- we construct reduced-order models for the emergent dynamics at different scales: (a) mesoscopic Integro-Partial Differential Equations (IPDEs); and (b) mean-field-type Stochastic Differential Equations (SDEs) embedded in a low-dimensional latent space, targeted to the neighborhood of the tipping point. We contrast the uses of the different models and the effort involved in learning them.Comment: 29 pages, 8 figures, 6 table