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Physics-based numerical models designed to study processes on the surface of the Earth are commonly built with conservation laws. Yet conservations laws are based on a treatable subset of all physical and biological processes operating in the system of interest, and empirical relationships are always required to fully describe systems (and provide mathematical “closure”). In contrast to conservation laws, empirical expressions used in model construction are inductive, and based on observations of phenomena. Any model that requires empirical expressions is also subject to revision as the empirical parameterization is refined: all empiricism is open to revision by corroborating or conflicting data. As more data becomes available, and more degrees of freedom are explored, it becomes harder to incorporate all available data into a single optimal empirical predictor. We argue in this contribution that empirical parameterizations for numerical models should be constructed using machine learning techniques because these techniques are built to operate on large, high dimensional datasets. Machine learning, in the context of this paper, defines a suite of algorithms used to develop predictive relationships (correlations) using a set of input data. Examples of commonly used machine learning techniques in the Earth sciences are artificial neural networks (e.g., Maier and Dandy, 2000; Pape et al., 2007; van Maanen et al., 2010), regression trees (e.g., Snelder et al., 2009; Oehler et al., 2012), Bayesian networks (e.g., Aguilera et al., 2011; Yates and Le Cozannet, 2012), and evolutionary algorithms (e.g., Knaapen and Hulscher, 2002; Ruessink, 2005; Goldstein et al., 2013). Machine learning techniques offer insight and are high-performance, reproducible, and scalable. The inclusion of powerful inductive techniques like those provided by machine learning offers the opportunity to enhance the predictions obtained from deductive approaches. Additionally the use of machine learning often leads to further insight. The use of empiricisms built from machine learning in a physics-based model results in a “hybrid” model. Combining the strengths of inductive (data-driven) and deductive (physics-based) approaches in a single hybrid model has been suggested as a valuable step forward in model development because of increases in accuracy (e.g., Babovic et al., 2001; Hall, 2004) and speed (Krasnopolsky and Fox-Rabinovitz, 2006). An additional benefit is the direct coupling of models to data,

Machine learning components in deterministic models: hybrid synergy in the age of data