Physics Constrained Nonlinear Regression Models for Time Series

A central issue in contemporary science is the development of data driven statistical nonlinear dynamical models for time series of partial observations of nature or a complex physical model. It has been established recently that adhoc quadratic multi-level regression models can have finite-time blow up of statistical solutions and/or pathological behavior of their invariant measure. Here a new class of physics constrained multi-level quadratic regression models are introduced, analyzed, and applied to build reduced stochastic models from data of nonlinear systems. These models have the advantages of incorporating memory effects in time as well as the nonlinear noise from energy conserving nonlinear interactions. The mathematical guidelines for the performance and behavior of these physics constrained multi-level regression models as well as filtering algorithms for their implementation are developed here. Data driven applications of these new multi-level nonlinear regression models are developed for test models involving a nonlinear oscillator with memory effects and the difficult test case of the truncated Burgers-Hopf (TBH) model. These new physics constrained quadratic multi-level regression models are proposed here as process models for Bayesian estimation through Markov Chain Monte Carlo algorithms of low frequency behavior in complex physical data. AMS classification scheme numbers: 93E11, 62M20, 93E11, 62L12, 65C20,62J02,62M10 Submitted to: Nonlinearity