Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uncertainty Quantification in Data-Driven Dynamical Models via Inverse Problem Solving

Mohamed Akrout, Dan Wilson

Abstract

Data-driven model identification strategies can be used to obtain phenomenological models that capture the temporal evolution of observable data. While it is usually straightforward to obtain such a model from time series data, for instance with least-squares fitting, it is generally difficult to quantify the uncertainty associated with the prediction of the temporal evolution of the observables. This paper considers a general framework for uncertainty quantification in data-driven dynamical models by framing prediction error through the lens of inverse problem theory. Building on Koopman-inspired model identification strategies that are suited for nonlinear dynamical models, we consider a prediction as an approximate measurement from which the original input state can be faithfully recovered, and define the prediction error as the MSE of solving the inverse problem that would yield this prediction. We demonstrate the efficacy of this approach on both numerical models and experimental data showing that it provides a robust uncertainty measure of model performance.

Uncertainty Quantification in Data-Driven Dynamical Models via Inverse Problem Solving

Abstract

Data-driven model identification strategies can be used to obtain phenomenological models that capture the temporal evolution of observable data. While it is usually straightforward to obtain such a model from time series data, for instance with least-squares fitting, it is generally difficult to quantify the uncertainty associated with the prediction of the temporal evolution of the observables. This paper considers a general framework for uncertainty quantification in data-driven dynamical models by framing prediction error through the lens of inverse problem theory. Building on Koopman-inspired model identification strategies that are suited for nonlinear dynamical models, we consider a prediction as an approximate measurement from which the original input state can be faithfully recovered, and define the prediction error as the MSE of solving the inverse problem that would yield this prediction. We demonstrate the efficacy of this approach on both numerical models and experimental data showing that it provides a robust uncertainty measure of model performance.
Paper Structure (18 sections, 36 equations, 9 figures)

This paper contains 18 sections, 36 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Koopman Operator Theory
  4. Data-Driven Model Identification Using Koopman Operator Theory
  5. Nonlinear Estimators for the Koopman Operator
  6. Inverse problems and expectation propagation
  7. Step 1: MMSE estimation of $\bm{x}^+$
  8. Step 2: Linear MMSE (LMMSE) estimation of $\bm{x}^-$
  9. Error Uncertainty Estimation via Inverse Problems
  10. Methodology
  11. Error quantification via Bayesian posterior variance
  12. The expectation propagation framework
  13. Variance as a metric for predictive confidence
  14. Results
  15. Conductance-Based Neural Model
  16. ...and 3 more sections

Figures (9)

  • Figure 1: VAMP's factor graph according to the factorization (\ref{['eq:VAMP_posterior-factorize']}).
  • Figure 2: Block diagram of the VAMP approach to solve the inverse problem with its two modules: one MMSE denoiser incorporating the prior $p(\bm{x}^+)$, and the LMMSE module. The two modules exchange extrinsic information/messages through the blocks.
  • Figure 3: Quantification of prediction error through Bayesian inverse reformulation and solving
  • Figure 4: Factor graph of the joint density factorization used in VAMP for error quantification.
  • Figure 5: Uncertainty quantification of the predictive performance for the neural model.
  • ...and 4 more figures