A Bayesian approach to estimating hidden variables as well as missing and wrong molecular interactions in ordinary differential equation-based mathematical models

Engelhardt, Benjamin; Kschischo, Maik; Fröhlich, Holger

doi:10.1098/rsif.2017.0332

Cited by 19 publications

(25 citation statements)

References 42 publications

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“…we did not assess the FISPO of the models used as case studies, since we did not have the tools for such analysis. Likewise, other related works have considered different instances of input reconstruction problems addressing the practical estimation problem [ 38 – 42 ]. In particular, Schelker et al [ 39 ] considered uncertainty in the input measurements within the general parameter estimation (PE) formulation, while Kaschek et al [ 38 ] used a calculus of variations-based approach and Trägårdh et al [ 40 ] formulated the input reconstruction as a Bayesian inference problem.…”

Section: Introductionmentioning

confidence: 99%

Full observability and estimation of unknown inputs, states and parameters of nonlinear biological models

Villaverde¹,

Tsiantis

Banga³

2019

J. R. Soc. Interface.

101

137

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In this paper, we address the system identification problem in the context of biological modelling. We present and demonstrate a methodology for (i) assessing the possibility of inferring the unknown quantities in a dynamic model and (ii) effectively estimating them from output data. We introduce the term Full Input-State-Parameter Observability (FISPO) analysis to refer to the simultaneous assessment of state, input and parameter observability (note that parameter observability is also known as identifiability). This type of analysis has often remained elusive in the presence of unmeasured inputs. The method proposed in this paper can be applied to a general class of nonlinear ordinary differential equations models. We apply this approach to three models from the recent literature. First, we determine whether it is theoretically possible to infer the states, parameters and inputs, taking only the model equations into account. When this analysis detects deficiencies, we reformulate the model to make it fully observable. Then we move to numerical scenarios and apply an optimization-based technique to estimate the states, parameters and inputs. The results demonstrate the feasibility of an integrated strategy for (i) analysing the theoretical possibility of determining the states, parameters and inputs to a system and (ii) solving the practical problem of actually estimating their values.

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Section: Introductionmentioning

confidence: 99%

Full observability and estimation of unknown inputs, states and parameters of nonlinear biological models

Villaverde¹,

Tsiantis

Banga³

2019

J. R. Soc. Interface.

101

137

View full text Add to dashboard Cite

show abstract

“…Reconstructing unknown inputs from outputs of open systems is useful in many settings. For modellers, the inputs provide important information about model errors and cues for model improvement or extension [12,13,34]. In biomedical systems, the unknown inputs can represent unmodelled environmental or physiological inputs, which might be interesting for the design of devices or measurement strategies.…”

Section: A Summary and Significance Of The Resultsmentioning

confidence: 99%

“…In electrical or secure networks, the unknown inputs could be attack signals, which need to be reconstructed and then mitigated. Unknown inputs can also be useful for improved state estimation [12][13][14] and data assimilation [53,77]. Thus, from the viewpoint of modellers and engineers, invertibility is a desirable prop- Table I with three different node selection schemes as a function of the number of inputs.…”

Section: A Summary and Significance Of The Resultsmentioning

confidence: 99%

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Structural Invertibility and Optimal Sensor Node Placement for Error and Input Reconstruction in Dynamic Systems

et al. 2019

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Despite recent progress in our understanding of complex dynamic networks, it remains challenging to devise sufficiently accurate models to observe, control or predict the state of real systems in biology, economics or other fields. A largely overlooked fact is that these systems are typically open and receive unknown inputs from their environment. A further fundamental obstacle are structural model errors caused by insufficient or inaccurate knowledge about the quantitative interactions in the real system.Here, we show that unknown inputs to open systems and model errors can be treated under the common framework of invertibility, which is a requirement for reconstructing these disturbances from output measurements. By exploiting the fact that invertibility can be decided from the influence graph of the system, we analyse the relationship between structural network properties and invertibility under different realistic scenarios. We show that sparsely connected scale free networks are the most difficult to invert. We introduce a new sensor node placement algorithm to select a minimum set of measurement positions in the network required for invertibility. This algorithm facilitates optimal experimental design for the reconstruction of inputs or model errors from output measurements. Our results have both fundamental and practical implications for nonlinear systems analysis, modelling and design.

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“…In this context, extended Kalman filtering [10], unscented Kalman filtering [11], and ensemble Kalman methods [12] have been applied as well. In addition, different methods have also been developed to address the issue of hidden variables and dynamics [13,14].…”

Section: Introductionmentioning

confidence: 99%

Systems biology informed deep learning for inferring parameters and hidden dynamics

Yazdani

Raissi

et al. 2019

Preprint

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Mathematical models of biological reactions at the system-level lead to a large set of ordinary differential equations with many unknown parameters that need to be inferred using relatively few experimental measurements. Having a reliable and robust algorithm for parameter inference and prediction of the hidden dynamics has been one of the core subjects in systems biology, and is the focus of this study. We have developed a novel systems-biology-informed deep learning algorithm that incorporates the system of ordinary differential equations into the neural networks. Enforcing these equations effectively adds constraints to the optimization procedure that manifests itself as an imposed structure on the observational data. Using few scattered and noisy measurements, we are able to infer the dynamics of unobserved species, systematic forcing and the unknown model parameters. We have successfully tested the algorithm for three different benchmark problems. Author summaryThe dynamics of systems biological processes are usually modeled using ordinary differential equations (ODEs), which introduce various unknown parameters that need to be estimated efficiently from noisy measurements of concentration for a few species only. In this work, we present a novel "systems-informed neural network" to infer the dynamics of experimentally unobserved species as well as the unknown parameters in the system of equations. By incorporating the system of ODEs into the neural networks we effectively add constraints to the optimization algorithm, which makes the method robust to measurement noise and few scattered observations. 2 holistic approach to deciphering the complexity of biological systems. To understand 3 the biological systems, we must understand the structures of the systems (both their 4 components and structural relationships), and their dynamics [1]. The dynamics of 5 systems biological processes are usually modeled using ordinary differential equations 6 (ODEs) that describe the time evolution of chemical and molecular species 7 concentrations. Once the pathway structure of chemical reactions is known, the 8 corresponding equations can be derived using widely accepted kinetic laws, such as the 9 law of mass action or the Michaelis-Menten kinetics [2]. 10 November 28, 2019 1/15Most system-level biological models introduce various unknown parameters, which 11 need to be estimated efficiently. Thus, the central challenge in computational modeling 12 of these systems could be the prediction of model parameters such as rate constants or 13 initial concentrations, and model trajectories such as time evolution of experimentally 14 unobserved concentrations. Due to the importance of parameter estimation, a lot of 15 attention has been given to this problem in the systems biology community. A lot of 16 research has been conducted on the applications of several optimization techniques, such 17 as linear and nonlinear least-squares fitting [3], genetic algorithms [4], and evolutionary 18 computation [5]. Considerable interest has a...

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A Bayesian approach to estimating hidden variables as well as missing and wrong molecular interactions in ordinary differential equation-based mathematical models

Cited by 19 publications

References 42 publications

Full observability and estimation of unknown inputs, states and parameters of nonlinear biological models

Full observability and estimation of unknown inputs, states and parameters of nonlinear biological models

Structural Invertibility and Optimal Sensor Node Placement for Error and Input Reconstruction in Dynamic Systems

Systems biology informed deep learning for inferring parameters and hidden dynamics

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