##### Abstract

This dissertation addresses the problem of the computer prediction of the approximate
behaviour of physical systems describable by ordinary differential equations.

Previous approaches to behavioural prediction have either focused on an exact
mathematical description or on a qualitative account. We advocate a middle ground: a
representation more coarse than an exact mathematical solution yet more specific than a
qualitative one. What is required is a mathematical expression, simpler than the exact
solution, whose qualitative features mirror those of the actual solution and whose
functional form captures the principal parameter relationships underlying the behaviour of
the real system. We term such a representation an approximate functional solution.

Approximate functional solutions are superior to qualitative descriptions because they
reveal specific functional relationships, restore a quantitative time scale to a process and
support more sophisticated comparative analysis queries. Moreover, they can be superior to
exact mathematical solutions by emphasizing comprehensibility, adequacy and practical
utility over precision.

Two strategies for constructing approximate functional solutions are proposed. The first
abstracts the original equation, predicts behaviour in the abstraction space and maps this
back to the approximate functional level. Specifically, analytic abduction exploits
qualitative simulation to predict the qualitative properties of the solution and uses this
knowledge to guide the selection of a parameterized trial function which is then tuned with
respect to the differential equation. In order to limit the complexity of a proposed
approximate functional solution, and hence maintain its comprehensibility,
back-of-the-envelope reasoning is used to simplify overly complex expressions in a
magnitude extreme. If no function is recognised which matches the predicted behaviour,
segment calculus is called upon to find a composite function built from known primitives
and a set of operators. At the very least, segment calculus identifies a plausible structure
for the form of the solution (e.g. that it is a composition of two unknown functions).
Equation parsing capitalizes on this partial information to look for a set of termwise
interactions which, when interpreted, expose a particular solution of the equation.

The second, and more direct, strategy for constructing an approximate functional solution is
embodied in the closed form approximation technique. This extends approximation
methods to equations which lack a closed form solution. This involves solving the
differential equation exactly, as an infinite series, and obtaining an approximate functional
solution by constructing a closed form function whose Taylor series is close to that of the
exact solution

The above techniques dovetail together to achieve a style of reasoning closer to that of an
engineer or physicist rather than a mathematician. The key difference being to sacrifice the
goal of finding the correct solution of the differential equation in favour of finding an
approximation which is adequate for the purpose to which the knowledge will be put.
Applications to Intelligent Tutoring and Design Support Systems are suggested.