Discovering the structure of partial differential equations from example behavior

Ljupco Todorovski, Saso Dzeroski, Ashwin Srinivasan, Jonathan Whiteley, David Gavaghan

Abstract

One of the most powerful and widely accepted analytical formalisms for modeling biological and physical systems is that of the partial differential equation (PDE). Establishing an acceptable PDE model for a dynamic system occupies a major portion of the work of the mathematical modeler. There are two main aspects to this activity. First, an appropriate structure has to be determined for the equations involved (the model identification problem). Second, acceptably accurate values for parameters are to be determined (the parameter estimation problem). Of these, the first is more challenging, and is the focus of this paper. We propose a method for discovering the structure of PDE models from example behavior. For simple PDE models, we illustrate that a straight-forward adaptation of existing equation discovery methods is sufficient. However, complex PDE models require a more sophisticated approach: a two-stage method is described in the paper. The efficacy of the approach is demonstrated initially by rediscovering the PDE models for several artificial problems. We also use it to obtain the structure of the classic FitzHugh-Nagumo model. This represents a very wide class of biological systems, making the model discovery method of interest to scientists concerned with the enterprise of obtaining a mathematical understanding of dynamic processes occurring in the life sciences.

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