Numerical models for simulating geophysical processes, such as ocean circulation or glacier flow, solve equations of motions. These equations consist of both fundamental conservation laws as well as constitutive laws. These constitutive laws are empirical relations that capture both material properties (sea water or glacier ice) and processes not resolved by the spatial discretization. Input parameters are therefore required to perform such simulations. These inputs are all uncertain and come in the form of model initial conditions, boundary conditions (such as atmospheric forcing), and parameters used in the constitutive laws (such as mixing, viscosity, or basal properties). Counting all of the parameters together (many of which are spatially varying and thus are two- or three-dimensional fields) leads to a very high-dimensional space of uncertain parameters that is typically greater than 10⁵. The model initialization problem consists of estimating optimal initial conditions (e.g., for forecasting), whereas the model calibration problem consists of inferring optimal model parameters, based on observations (akin to neural network training in the now popular language of machine learning). In both cases one seeks compute the sensitivity of an objective function (or quantity of interest) to all uncertain inputs, or, in mathematical terms, the gradient of the objective functions with respect to the high-dimensional space of input parameters (often referred to as control variables). Computing this gradient through direct (or forward) approaches, such as finite-differencing is expensive, if not computationally impractical.

The adjoint method is a powerful approach to efficiently and comprehensively compute such sensitivities. This method enables efficient ocean data assimilation and sensitivity analysis, and thus has widespread impact. Beyond geosciences, disciplines impacted include all fields where prior models inform the research by revealing the sensitivity of these models to inputs. Examples include optimization of tidal turbine arrays, airfoil design, finance, and epidemiological modeling. Writing an adjoint model from scratch is as complex as writing its nonlinear parent model (such as an ocean circulation model or ice flow model). Automatic or algorithmic differentiation (AD) provides a powerful tool for generating the adjoint model in an automated manner, based on the nonlinear parent model. It is ubiquitous in machine learning (where the adjoint method is known as “backpropagation”), but so far less widespread in geoscientific modeling. Although powerful conceptually, automatic generation of reliable, efficient adjoint code via AD remains a complex process. 

The major goal of this project is to remove some of the hurdles that have prevented the wider use of adjoint methods through maturing an open-source automatic differentiation software, OpenAD. Moreover, we aimed to make the tools more user friendly, and to educate on their use via tutorial examples, presentations, workshops, and peer-reviewed publications demonstrating the utility of the methods. Project success was achieved by maturing and presenting the software (Intellectual Merit), but also via supporting the mentorship of multiple students, including REUs, and postdoctoral researchers (Broader Impact). The student and postdoc work led to conference presentations and peer-reviewed publications. 

Three specific example outcomes were (i) the generation of an adjoint model of the SImulation COde for POLythermal Ice Sheets (SICOPOLIS) to conduct sensitivity studies of the Greenland and Antarctic ice sheets; (ii) generation of the adjoint of an ocean model and an ice stream model of the MIT general circulation model (MITgcm) for simulating coupled ice stream-ice shelf-ocean interactions in West Antarctica; and (iii) improving uncertainty quantification via adjoint derivative-based methods to inform biogeochemical observing system design. 

This project has enabled a larger fraction of the community to generate and utilize adjoint models for sensitivity analyses, state estimation parameter calibration, or model initialization. Enabling a broad community of researchers to efficiently probe the sensitivity of domain-specific quantities of interest will allow impactful progress in understanding pressing problems that our society faces.

Last Modified: 01/06/2021
Modified by: Patrick Heimbach