Learning dynamic Boltzmann distributions as reduced models of spatial chemical kinetics.

  title={Learning dynamic Boltzmann distributions as reduced models of spatial chemical kinetics.},
  author={Oliver K. Ernst and Thomas M. Bartol and Terrence J. Sejnowski and Eric Mjolsness},
  journal={The Journal of chemical physics},
  volume={149 3},
Finding reduced models of spatially distributed chemical reaction networks requires an estimation of which effective dynamics are relevant. We propose a machine learning approach to this coarse graining problem, where a maximum entropy approximation is constructed that evolves slowly in time. The dynamical model governing the approximation is expressed as a functional, allowing a general treatment of spatial interactions. In contrast to typical machine learning approaches which estimate the… 
Deep Learning Moment Closure Approximations using Dynamic Boltzmann Distributions
This work has developed a method to learn moment closure approximations directly from data using dynamic Boltzmann distributions (DBDs), which can be applied broadly to learn deep generative models in applications where infinite systems of differential equations arise.
TMI: Thermodynamic inference of data manifolds
TMI simultaneously learns from data intensive and extensive variables and achieves dimensionality reduction through a multiplicative, positive valued, and interpretable decomposition of the data.
Prospects for Declarative Mathematical Modeling of Complex Biological Systems
  • E. Mjolsness
  • Computer Science
    Bulletin of mathematical biology
  • 2019
The operator algebra semantics of an increasingly powerful series of declarative modeling languages including reaction-like dynamics of parameterized and extended objects are defined and an outline of a “meta-hierarchy” for organizingDeclarative models and the mathematical methods that can fruitfully manipulate them is outlined.
A high-bias, low-variance introduction to Machine Learning for physicists
Machine learning dynamic correlation in chemical kinetics.
It is shown that machine learning (ML) can be used to construct accurate moment closures in chemical kinetics using the lattice Lotka-Volterra model as a model system and MLMC is a promising tool to interpolate KMC simulations or construct pretrained closures that would enable researchers to extract useful insight at a fraction of the computational cost.
Learning moment closure in reaction-diffusion systems with spatial dynamic Boltzmann distributions.
A lattice version of the Rössler chaotic oscillator is studied, which illustrates the accuracy of the moment closure approximation made by the machine-learning approach to model reduction based on the Boltzmann machine and its dimensionality reduction power.


Model reduction for stochastic CaMKII reaction kinetics in synapses by graph-constrained correlation dynamics
The novel model reduction method, ‘graph-constrained correlation dynamics’, requires a graph of plausible state variables and interactions as input and parametrically optimizes a set of constant coefficients appearing in differential equations governing the time-varying interaction parameters that determine all correlations between variables in the reduced model at any time slice.
Time-Ordered Product Expansions for Computational Stochastic Systems Biology
The time-ordered product framework of quantum field theory can be used systematically to derive simulation and parameter-fitting algorithms for stochastic systems and can be interpreted in terms of Feynman diagrams.
Exact Stochastic Simulation of Coupled Chemical Reactions
There are two formalisms for mathematically describing the time behavior of a spatially homogeneous chemical system: The deterministic approach regards the time evolution as a continuous, wholly
A closure scheme for chemical master equations
A closure scheme is presented that solves the master probability equation of networks of chemical or biochemical reactions, and a most striking finding is that the steady state of stochastic reaction networks can now be readily computed in a single-step calculation, without the need to simulate the evolution of the probability distribution in time.
Fast Monte Carlo Simulation Methods for Biological Reaction-Diffusion Systems in Solution and on Surfaces
It is shown how spatially realistic Monte Carlo simulations of biological systems can be far more cost-effective than often is assumed, and provide a level of accuracy and insight beyond that of continuum methods.
Mathematics of small stochastic reaction networks: a boundary layer theory for eigenstate analysis.
A novel way of finding the eigenstates of this system of difference-differential equations, using perturbation analysis of ordinary differential equations arising from approximation of the difference equations is presented.
Perspective: Stochastic algorithms for chemical kinetics.
The perspective on stochastic chemical kinetics is outlined, paying particular attention to numerical simulation algorithms, and some of the more promising strategies for dealing stochastically with stiff systems, rare events, and sensitivity analysis are taken.
Correlations in stochastic theories of chemical reactions
A comprehensive study of correlations in linear and nonlinear chemical reactions is presented using coupled chemical and diffusion master equations. As a consequence of including correlations in
A Machine Learning Method for the Prediction of Receptor Activation in the Simulation of Synapses
This work has developed a machine-learning based method that can accurately predict relevant aspects of the behavior of synapses, such as the percentage of open synaptic receptors as a function of time since the release of the neurotransmitter, with considerably lower computational cost compared with the conventional Monte Carlo alternative.
The Importance of Being Discrete (and Spatial)
Abstract We consider and compare four approaches to modeling the dynamics of spatially distributed systems: mean field approaches (described by ordinary differential equations) in which every