Model Reduction and Effective Theories in Physics, Biology, and Beyond

Dr. Mark Transtrum
Department of Physics and Astronomy
Brigham Young University

The success of science is due in large part to the hierarchical nature of physical theories. These effective theories model natural phenomena as if the physics at macroscopic length scales were almost independent of the underlying, shorter-length-scale details. The efficacy of these simplified models can be understood in terms of parameter identifiability.  Parameters associated with microscopic degrees of freedom are usually unidentifiable as quantified by the Fisher Information Matrix.  I apply an information geometric approach in which a microscopic, mechanistic model is interpreted as a manifold of predictions in data space.  Model manifolds are often characterized by a hierarchy of boundaries--faces, edges, corners, hyper-corners, etc. These boundaries correspond to reduced-order models, leading to a model reduction technique known as the Manifold Boundary Approximation Method.  In this way, effective models can be systematically derived from microscopic first principles for a variety of complex systems in physics, biology, and other fields.