**2. The cost of problem formulation with mathematical models**

Unfortunately, model formulation is a rather strenuous task performed in the creative mind of a modeler. The graphs from Aris' book in **Figure 2** illustrate profound modeling insights generated about reactive systems. The wonderful illustrations were not created spontaneously, but were conceived by intensive contemplation. From a practical point of view, it is legitimate to ask how taxing is the art of model generation, and whether industry can afford to pay for the labor of modeling artists? Let me offer an empirical estimate for the cost of mathematical model generation using examples of my past 20 years in chemical engineering research. My PhD thesis on coal gasification mathematical problem formulation took 2 years: its resulting nonlinear equations were encoded in Fortran. Today, this task may be accomplished a few months faster using modern modeling languages such as gPROMS or GAMS [2, 3]. My second example concerns chemical process flowsheets typically analyzed in

#### **Figure 2.**

*Collage of Aris' figures about reactive systems. Assembled from Aris' book entitled mathematical modeling techniques [1].*

*Technology, Science and Culture - A Global Vision*

era of computing.

The replacement of theory-less domain by another for which the theory is known has hardly ever been more successfully demonstrated than in the revolutionary discoveries of Alan Turing whose model of mathematical operations establishes the logical basis of modern computing. Turing realized that algebraic operations in our decimal system, additions and multiplications in our common natural number system, can be replaced with AND or OR logic operators within a binary number system. Accordingly, natural and real numbers could be digitized. More importantly, expensive tasks such as evaluating long lists of summation or multiplication operations could be executed by clocking thousands of AND and OR operations on

His contribution fits the definition of modeling as the replacement of decimal operations in the common number systems with the extremely simple and fast binary AND/OR logic operations; his breakthrough achievement rang in the digital age. His seminal work rooted in an ingenious mathematical model that combined well-known facts of mathematical logic with electronic principles, heralded a new

Discoveries like Turing's spring from inspiration and creativity; thus, mathematical modeling is an art. But is there a formal way to generate modeling art? The sciences offer a useful template for model-based discovery and learning. I propose an analogy with a feedback circuit depicted in **Figure 1**. The learning cycle begins with a problem formulation conceived in the mind of the modeler that described properties and possible transformations about a novel physical prototype such as a chemical process, whose extent and critical parameters may not yet be fully known or understood. Problem formulation implies the transformation of domain-specific states and transition into suitable mathematical relations. When the physical domain, its state descriptions, and transitions are adequately mapped into a suitable mathematical surrogate, the algorithmic machinery is invoked to make mathematical predictions about the system states, X, typically using digital computers. Mathematical analysis involves the solution of mathematical equations for which we typically use computers. This involves the numerical solution of algebraic equations, nonlinear models, and optimization or system dynamics using well-known rules in the mathematical domain. But model-based learning does not stop here: properties of the mathematical surrogate, X, are not the object of study. Instead, it is critical to interpret (=or back-transform) predictions in terms of the physical states in the problem domain. In engineering, this means inferences about the states of the matter such as pressure, temperatures, and concentrations (P, T, and C). Since the range and deliberate manipulation in the original prototype space are not fully known, or may be just vaguely known, it is often necessary to sharpen the original mathematical problem formulation or assumptions. The incorporation of feedback about the original

a simple electronic machine, which became later known as a *computer*.

**24**

**Figure 1.**

*Schematic of the model-based learning paradigm.*

**Figure 3.**

*Overview of the cost of modeling. Problem formulation absorbs more than 90% of the cost for the development of mathematical models.*

chemical engineering senior design course. Development and analysis of the process flowsheet may take a student team operating the Aspen FlowSheet Software at the beginner's level about 45 man-hours, or 3 hours per week for 15 weeks. An expert user may be able to set up the flowsheet in only 10 hours. The third example concerns the development of a computational fluid mechanics (CFD) model, of which we will see more later in the study of the brain. Using commercial CFD software such as FLUENT [4], it may take a few days or weeks to set up a routine flow problem such as laminar flow in a cylindrical domain. A complex problem like subject-specific blood flow predictions such as an aneurysm may require 1–2 years of a PhD student. In biological systems, some CFD problems require even more than 2 years. The chart of **Figure 3** shows that in cost for model formulation falls in the range of a few hundred thousand dollars for junior level engineers; it may reach or exceed a half a million dollars if the modeling problem requires an expert such as a senior expert or a scientist. In contrast, the computational time for merely solving the mathematical model was calculated to amount to only \$500 of CPU time (=1 week cpu time). Accordingly, the cost of formulating models is much higher than the expense for solving mathematical equations. The expense for mathematical model-based learning stems mainly from the effort for problem formulation, a much smaller fraction is attributable to the solution on the computer. In a computationally intense scenario (=1 week CPU time), computational cost may accumulate about 10% of the cost. In the more complex modeling situation, the cost essentially lies mainly in the mathematical model formulation, interpretation, and testing of the system. It is therefore desirable to accelerate model generation and thus reduce its cost.
