31 research outputs found
ODE parameter inference using adaptive gradient matching with Gaussian processes
Parameter inference in mechanistic models based on systems of coupled differential equa- tions is a topical yet computationally chal- lenging problem, due to the need to fol- low each parameter adaptation with a nu- merical integration of the differential equa- tions. Techniques based on gradient match- ing, which aim to minimize the discrepancy between the slope of a data interpolant and the derivatives predicted from the differen- tial equations, offer a computationally ap- pealing shortcut to the inference problem. The present paper discusses a method based on nonparametric Bayesian statistics with Gaussian processes due to Calderhead et al. (2008), and shows how inference in this model can be substantially improved by consistently inferring all parameters from the joint dis- tribution. We demonstrate the efficiency of our adaptive gradient matching technique on three benchmark systems, and perform a de- tailed comparison with the method in Calder- head et al. (2008) and the explicit ODE inte- gration approach, both in terms of parameter inference accuracy and in terms of computa- tional efficiency
On solving Ordinary Differential Equations using Gaussian Processes
We describe a set of Gaussian Process based approaches that can be used to
solve non-linear Ordinary Differential Equations. We suggest an explicit
probabilistic solver and two implicit methods, one analogous to Picard
iteration and the other to gradient matching. All methods have greater accuracy
than previously suggested Gaussian Process approaches. We also suggest a
general approach that can yield error estimates from any standard ODE solver
Computational inference in systems biology
Parameter inference in mathematical models of biological pathways, expressed as coupled ordinary differential equations (ODEs), is a challenging problem. The computational costs associated with repeatedly solving the ODEs are often high. Aimed at reducing this cost, new concepts using gradient matching have been proposed. This paper combines current adaptive gradient matching approaches, using Gaussian processes, with a parallel tempering scheme, and conducts a comparative evaluation with current methods used for parameter inference in ODEs
Fast inference in nonlinear dynamical systems using gradient matching
Parameter inference in mechanistic models of
coupled differential equations is a topical problem.
We propose a new method based on kernel
ridge regression and gradient matching, and
an objective function that simultaneously encourages
goodness of fit and penalises inconsistencies
with the differential equations. Fast minimisation
is achieved by exploiting partial convexity
inherent in this function, and setting up an iterative
algorithm in the vein of the EM algorithm.
An evaluation of the proposed method on various
benchmark data suggests that it compares
favourably with state-of-the-art alternatives
Controversy in mechanistic modelling with Gaussian processes
Parameter inference in mechanistic models based on non-affine differential equations is computationally onerous, and various faster alternatives based on gradient matching have been proposed. A particularly promising approach is based on nonparametric Bayesian modelling with Gaussian processes, which exploits the fact that a Gaussian process is closed under differentiation. However, two alternative paradigms have been proposed. The first paradigm, proposed at NIPS 2008 and AISTATS 2013, is based on a product of experts approach and a marginalization over the derivatives of the state variables. The second paradigm, proposed at ICML 2014, is based on a probabilistic generative model and a marginalization over the state variables. The claim has been made that this leads to better inference results. In the present article, we offer a new interpretation of the second paradigm, which highlights the underlying assumptions, approximations and limitations. In particular, we show that the second paradigm suffers from an intrinsic identifiability problem, which the first paradigm is not affected by
常微分方程式に従う時間依存混合モデル
ISM Online Open House, 2020.10.27統計数理研究所オープンハウス(オンライン開催)、R2.10.27ポスター発
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