Learning Differential Equations in the Presence of Data and Model Uncertainty

ABSTRACT. Models of physical systems often have elusive constitutive details which must be determined empirically. A new method, approximate model inference (AMI), is proposed for data-driven parameter estimation in differential equations where uncertainties exist in the model itself. This is formulated in the context of likelihood maximization over both the state and parameter variables. This approach is shown to deal with both significant noise and incomplete data. Practical implementation and optimization strategies are discussed both for systems of ordinary differential and partial differential equations. Experiments using synthetic data are performed for a variety of test problems, including those exhibiting chaotic or complex spatiotemporal behavior. The resulting estimates are shown to be competitive and often better than standard regression approaches