ANN technology, a human “learning” paradigm form of artificial intelligence, is a compelling and often superior alternative to physical-based and statistical modeling approaches. An ANN, through proper development and training, “learns” the system behavior of interest by processing representative data patterns through its architecture.  One of the powerful features of ANN technology is that, as more data becomes available, ANN models can easily be updated and improved with additional training to more fully capture subtle tendencies.  They have been proven to outperform other advanced modeling techniques in a variety of applications, and are used extensively in many sectors, including Wall Street, the military, and NASA. 

Unlike a physical-based (e.g. numerical) model, ANNs do not rely upon governing physical laws, so difficult to estimate parameters (e.g. hydraulic conductivity, streambed thickness, etc.) are typically not required for their development and operation. Instead, more easily measurable and less uncertain variables like water levels and air temperature can be used as inputs or predictor variables.  Unlike physical-based and statistical models, ANNs are not constrained by simplifying mathematical assumptions (e.g. linear system, normal distribution, etc.) or physical assumptions (e.g. laminar flow).  Because of their powerful non-linear modeling capability (see below), ANNs can accurately model highly non-linear and complex phenomena. In addition, unlike numerical models, ANNs can easily be initialized to real-time conditions, improving short-term predictive accuracy.  

Selection of an appropriate set of input or predictor variables during initial ANN model development requires a basic understanding of the governing system dynamics.  One of the great advantages of ANN models is they can often use more easily measurable “surrogate” variables (e.g. temperature and precipitation) in lieu of difficult to estimate parameters typically required by physical-based models (e.g. areal recharge).  The ANN-provided sensitivity analysis in conjunction with trial and error iterations can help the modeler converge to the most appropriate feasible set of predictor variables.

Robust ANN development is dependent upon the quantity and quality of the data used to train the models. The appropriate training set size or the number of events used for ANN learning depends upon a number of factors, including the required ANN accuracy, the probability distribution of behavior, the level of noise in the system, the complexity of the system, and the size of the ANN model (i.e. number of nodes).  Because many hydrologic and energy systems are relatively “well-behaved”, where small changes in input values do not produce significantly different or even contradictory output values, relatively few data patterns are often necessary.  That said, the range of data should ideally span the expected range of system behavior or performance. 

Each ANN-derived state-transition equation expresses the output variable(s) explicitly in terms of the known input variables and connection weights formed during learning (i.e. training).  Even when there are multiple state-transition equations used to simultaneously predict common output variables (e.g. water levels), because each state-transition equation is independent of the others, only simple arithmetic operations are required to solve for the unknown output variable(s). This computational ease of solution differs greatly from that of numerical models, which must use advanced numerical algorithms to simultaneously solve their set of dependent equations.  In addition, the condensed nature of the ANN approach can result in state-transition equations that number orders of magnitude less than the equations constituting a typical numerical model.  By comparison then, solving the ANN-derived state-transition equations is often significantly more efficient and fast. Consequently, this approach can be used to simulate large numbers of possible scenarios that might otherwise be infeasible with a large numerical groundwater model.

Mathematical optimization formulates complex management problem within a logical and transparent mathematical structure that can be solved using a variety of rigorous optimization algorithms.  The optimization formulation consists of an objective function and constraint set, expressed in terms of the decision variables for which the optimal values are unknown. The optimization program computes the optimal values, and includes a sensitivity analyses of how the optimal solution changes with different constraint bounds and coefficients.  Typically, “optimal solutions” are identified by a trial and error approach by systematically varying the values of the decision variables, which is not only highly inefficient and time consuming, but may not result in identification of even a good solution.  In contrast, optimization algorithms converge to local (i.e. non-linear problems) if not global (i.e. linear problems) optima quickly and efficiently. In addition, multiobjective optimization generates the formal trade-off curve for multiple and even conflicting objectives, from which the optimal compromise solution can be identified using a variety of methods. 

The ANN approach not only offers a computationally efficient and numerically stable optimization alternative, but can provide superior solutions.  Using ANNs can help avoid identification of erroneous solutions, which can occur when the commonly used response coefficient methodology is applied to non-linear (e.g. unconfined aquifer) optimization problems.  For example, Riefler and Ahlfeld (1996) found that perturbation values for an unconfined problem that are either too large or too small can produce an erroneous solution.  To avoid these problems, the generally less efficient embedding optimization approach can be used, where the simulation model is embedded into the optimization formulation as constraints.  The ANN derived state-transition equations, used in lieu of a numerical model, can reduce the number of equations in the constraint set by orders of magnitude. Fewer mathematical operations are then required during optimization, not only increasing computational efficiency, but also minimizing round-off and precision errors that result from large numbers of mathematical operations (Szidarovszky and Yakowitz, 1978).  In addition, because ANN models can achieve superior accuracy, exceeding that of physical-based models, the computed optimal solution by extension, is more accurate.

• Provide superior real-time predictive accuracy that exceeds state of the art numerical models.
• Can utilize continuous data streams in real-time, as ANN are “data-driven” models that excel with huge data sets, and are ideally suited for SCADA-type data collection systems.
• Can easily be initialized to real-time conditions, increasing predictive accuracy and providing solutions that reflect existing system states.
• Their condensed and efficient mathematical form typically allows computation of alternative management solutions in a few seconds or less.
• Their condensed and efficient mathematical form is ideal for performing formal optimization, where round-off errors and/or perturbation problems associated when using physical-based numerical models.
• Their superior prediction accuracy results in computation of more accurate optimal solutions.
• Provide valuable insights between cause and effect relationships, improving understanding of system dynamics.
• Can be used to improve data collection strategies by identifying important variables that influence system behavior of interest.
• Can easily be combined with physical-based equations and/or interpolation methods to increase the domain of predictions and optimization.