Learning unknown ODE models with Gaussian processes

In conventional ODE modelling coefficients of an equation driving the system state forward in time are estimated. However, for many complex systems it is practically impossible to determine the equations or interactions governing the underlying dynamics. In these settings, parametric ODE model cannot be formulated. Here, we overcome this issue by introducing a novel paradigm of nonparametric ODE modelling that can learn the underlying dynamics of arbitrary continuous-time systems without prior knowledge. We propose to learn non-linear, unknown differential functions from state observations using Gaussian process vector fields within the exact ODE formalism. We demonstrate the model's capabilities to infer dynamics from sparse data and to simulate the system forward into future.Comment: 11 pages, 2 page appendi

On Stochastic Differential Equations: Theory and Biochemical Applications

The time evolution of chemical systems is traditionally modeled using deterministic ordinary differential equations. The deterministic approach, in general, describes the average time series behavior of the system, but is incapable of capturing the random nature of chemical reactions. Thus, in order to be able to construct physically realistic models, stochastic methods have to be used. This is especially the case in many biochemical applications in which the reaction volume is small, the molecular concentrations are low, and the impact of random fluctuations is significant. Biochemical reactions can be modeled stochastically using numerous different approaches. For instance, stochastic differential equations have been proposed as a promising method to model biochemical reactions stochastically. The main goal of this Master of Science Thesis is to study the theory behind stochastic differential equations and to apply the theory to the modeling of biochemical systems. First, probability theory and stochastic processes are studied from the measure-theoretic point of view. Second, fundamental definitions and results from the field of stochastic calculus are presented. Applications based on the theory will then follow. In the simulation part, two biochemical systems, the chemical degradation and the Lotka reactions, are modeled by means of stochastic differential equations. The simulation of Lotka reactions proves the excellence of this stochastic approach. The stochastic differential equation model takes the natural fluctuations into account, and is thus capable of describing the dynamical properties of the system, even in the case, in which the traditional deterministic model fails to capture the temporal behavior. Asiasanat: stochastic differential equations, Ito calculus, modeling biochemical system

On Stochastic Differential Equations: Theory and Biochemical Applications

The time evolution of chemical systems is traditionally modeled using deterministic ordinary differential equations. The deterministic approach, in general, describes the average time series behavior of the system, but is incapable of capturing the random nature of chemical reactions. Thus, in order to be able to construct physically realistic models, stochastic methods have to be used. This is especially the case in many biochemical applications in which the reaction volume is small, the molecular concentrations are low, and the impact of random fluctuations is significant. Biochemical reactions can be modeled stochastically using numerous different approaches. For instance, stochastic differential equations have been proposed as a promising method to model biochemical reactions stochastically. The main goal of this Master of Science Thesis is to study the theory behind stochastic differential equations and to apply the theory to the modeling of biochemical systems. First, probability theory and stochastic processes are studied from the measure-theoretic point of view. Second, fundamental definitions and results from the field of stochastic calculus are presented. Applications based on the theory will then follow. In the simulation part, two biochemical systems, the chemical degradation and the Lotka reactions, are modeled by means of stochastic differential equations. The simulation of Lotka reactions proves the excellence of this stochastic approach. The stochastic differential equation model takes the natural fluctuations into account, and is thus capable of describing the dynamical properties of the system, even in the case, in which the traditional deterministic model fails to capture the temporal behavior. Asiasanat: stochastic differential equations, Ito calculus, modeling biochemical system

Data-driven mechanistic analysis method to reveal dynamically evolving regulatory networks

Motivation: Mechanistic models based on ordinary differential equations provide powerful and accurate means to describe the dynamics of molecular machinery which orchestrates gene regulation. When combined with appropriate statistical techniques, mechanistic models can be calibrated using experimental data and, in many cases, also the model structure can be inferred from time-course measurements. However, existing mechanistic models are limited in the sense that they rely on the assumption of static network structure and cannot be applied when transient phenomena affect, or rewire, the network structure. In the context of gene regulatory network inference, network rewiring results from the net impact of possible unobserved transient phenomena such as changes in signaling pathway activities or epigenome, which are generally difficult, but important, to account for. Results: We introduce a novel method that can be used to infer dynamically evolving regulatory networks from time-course data. Our method is based on the notion that all mechanistic ordinary differential equation models can be coupled with a latent process that approximates the network structure rewiring process. We illustrate the performance of the method using simulated data and, further, we apply the method to study the regulatory interactions during T helper 17 (Th17) cell differentiation using time-course RNA sequencing data. The computational experiments with the real data show that our method is capable of capturing the experimentally verified rewiring effects of the core Th17 regulatory network. We predict Th17 lineage specific subnetworks that are activated sequentially and control the differentiation process in an overlapping manner.Peer reviewe

Learning unknown ODE models with Gaussian processes

In conventional ODE modelling coefficients of an equation driving the system state forward in time are estimated. However, for many complex systems it is practically impossible to determine the equations or interactions governing the underlying dynamics. In these settings, parametric ODE model cannot be formulated. Here, we overcome this issue by introducing a novel paradigm of nonparametric ODE modelling that can learn the underlying dynamics of arbitrary continuous-time systems without prior knowledge. We propose to learn non-linear, unknown differential functions from state observations using Gaussian process vector fields within the exact ODE formalism. We demonstrate the model’s capabilities to infer dynamics from sparse data and to simulate the system forward into future.Peer reviewe

Learning unknown ODE models with Gaussian processes

In conventional ODE modelling coefficients of an equation driving the system state forward in time are estimated. However, for many complex systems it is practically impossible to determine the equations or interactions governing the underlying dynamics. In these settings, parametric ODE model cannot be formulated. Here, we overcome this issue by introducing a novel paradigm of nonparametric ODE modelling that can learn the underlying dynamics of arbitrary continuous-time systems without prior knowledge. We propose to learn non-linear, unknown differential functions from state observations using Gaussian process vector fields within the exact ODE formalism. We demonstrate the model’s capabilities to infer dynamics from sparse data and to simulate the system forward into future.Peer reviewe