3 research outputs found
Prospects for Declarative Mathematical Modeling of Complex Biological Systems
Declarative modeling uses symbolic expressions to represent models. With such
expressions one can formalize high-level mathematical computations on models
that would be difficult or impossible to perform directly on a lower-level
simulation program, in a general-purpose programming language. Examples of such
computations on models include model analysis, relatively general-purpose
model-reduction maps, and the initial phases of model implementation, all of
which should preserve or approximate the mathematical semantics of a complex
biological model. The potential advantages are particularly relevant in the
case of developmental modeling, wherein complex spatial structures exhibit
dynamics at molecular, cellular, and organogenic levels to relate genotype to
multicellular phenotype. Multiscale modeling can benefit from both the
expressive power of declarative modeling languages and the application of model
reduction methods to link models across scale. Based on previous work, here we
define declarative modeling of complex biological systems by defining the
operator algebra semantics of an increasingly powerful series of declarative
modeling languages including reaction-like dynamics of parameterized and
extended objects; we define semantics-preserving implementation and
semantics-approximating model reduction transformations; and we outline a
"meta-hierarchy" for organizing declarative models and the mathematical methods
that can fruitfully manipulate them
Learning Moment Closure in Reaction-Diffusion Systems with Spatial Dynamic Boltzmann Distributions
Many physical systems are described by probability distributions that evolve
in both time and space. Modeling these systems is often challenging to due
large state space and analytically intractable or computationally expensive
dynamics. To address these problems, we study a machine learning approach to
model reduction based on the Boltzmann machine. Given the form of the reduced
model Boltzmann distribution, we introduce an autonomous differential equation
system for the interactions appearing in the energy function. The reduced model
can treat systems in continuous space (described by continuous random
variables), for which we formulate a variational learning problem using the
adjoint method for the right hand sides of the differential equations. This
approach allows a physical model for the reduced system to be enforced by a
suitable parameterization of the differential equations. In this work, the
parameterization we employ uses the basis functions from finite element
methods, which can be used to model any physical system. One application domain
for such physics-informed learning algorithms is to modeling reaction-diffusion
systems. We study a lattice version of the R{\"o}ssler chaotic oscillator,
which illustrates the accuracy of the moment closure approximation made by the
method, and its dimensionality reduction power