Adjoint-Matching Neural Network Surrogates for Fast 4D-Var Data Assimilation

The data assimilation procedures used in many operational numerical weather forecasting systems are based around variants of the 4D-Var algorithm. The cost of solving the 4D-Var problem is dominated by the cost of forward and adjoint evaluations of the physical model. This motivates their substitution by fast, approximate surrogate models. Neural networks offer a promising approach for the data-driven creation of surrogate models. The accuracy of the surrogate 4D-Var problem's solution has been shown to depend explicitly on accurate modeling of the forward and adjoint for other surrogate modeling approaches and in the general nonlinear setting. We formulate and analyze several approaches to incorporating derivative information into the construction of neural network surrogates. The resulting networks are tested on out of training set data and in a sequential data assimilation setting on the Lorenz-63 system. Two methods demonstrate superior performance when compared with a surrogate network trained without adjoint information, showing the benefit of incorporating adjoint information into the training process.

Investigation of Nonlinear Model Order Reduction of the Quasigeostrophic Equations through a Physics-Informed Convolutional Autoencoder

Reduced order modeling (ROM) is a field of techniques that approximates complex physics-based models of real-world processes by inexpensive surrogates that capture important dynamical characteristics with a smaller number of degrees of freedom. Traditional ROM techniques such as proper orthogonal decomposition (POD) focus on linear projections of the dynamics onto a set of spectral features. In this paper we explore the construction of ROM using autoencoders (AE) that perform nonlinear projections of the system dynamics onto a low dimensional manifold learned from data. The approach uses convolutional neural networks (CNN) to learn spatial features as opposed to spectral, and utilize a physics informed (PI) cost function in order to capture temporal features as well. Our investigation using the quasi-geostrophic equations reveals that while the PI cost function helps with spatial reconstruction, spatial features are less powerful than spectral features, and that construction of ROMs through machine learning-based methods requires significant investigation into novel non-standard methodologies.