Building Digital Twins of Cities Via Governing Equations

Nonlinearity of interactions often gives rise to complex spatiotemporal structures and processes, characterizing urban evolution and self-organization with facilitating chaotic and non-chaotic transitions in space and time. This chapter deals with the learning of reduced-order models from high-dimensional approximations of urban dynamics with known governing equations. Using land use transition as an example, it shows the way to learn the reduced-order model of the high-dimensional model by projecting it onto a low dimensional manifold, a low-dimensional subspace. It seeks the intrinsic modes which constitute the solution space of the dynamical system and preserve the original complexity of land use dynamics. Using truncated singular value decomposition, we obtain the proper orthogonal decomposition modes intrinsic to the dynamical system that form a subspace that optimally represents the data containing the original dynamics. Applying to urban population growth with and without migration, the chapter also demonstrates how reduced-order models of parameterized high-dimensional models can be obtained through low-dimensional manifold learning. Furthermore, it discusses how to leverage sparsity with regularization to learn a parsimonious global basis that expand the parameterized space of the underlying dynamics governed by known equations.