Investigating Model Form Error Estimation for Sparse Data
Kyle D. Neal
1
, Mohammad Khalil
2
, and Teresa Portone
3
1,3
Sandia National Laboratories, Albuquerque, NM, 87123, USA
kneal@sandia.gov
tporton@academic.edu
2
Sandia National Laboratories, Livermore, CA, 94550, USA
mkhalil@sandia.gov
ABSTRACT
Computational simulations of dynamical systems often in-
volve the use of mathematical models and algorithms to
mimic and analyze complex real-world phenomena. By lever-
aging computational power, simulations enable researchers
to explore and understand systems that are otherwise chal-
lenging to study experimentally. They offer a cost-effective
and efficient means to predict and analyze the behavior of en-
gineering, biological, and social systems. However, model
form error arises in computational simulations from simpli-
fications, assumptions, and limitations inherent in the math-
ematical model formulation. Several methods for address-
ing model form error have been proposed in the literature,
but their robustness in the face of challenges inherent to real-
world systems has not been thoroughly investigated. In this
work, a data assimilation-based approach for model form er-
ror estimation is investigated in the presence of sparse ob-
servation data. An extension for including physics-based do-
main knowledge to improve estimation performance is pro-
posed. A computational simulation based on the Lotka-
Volterra equations is used for demonstration.
1. INTRODU CTION
Model form error (MFE) is a significant challenge in compu-
tational simulations, where mathematical models are used to
represent complex physical phenomena. It refers to the differ-
ence between the mathematical model and the true behavior
of the system being simulated. This error can arise from var-
ious sources, such as neglecting certain physical phenomena,
using simplified mathematical equations, or making assump-
tions about parameter values. MFE is a fundamental aspect
of the model itself and is independent of any specific data or
observations. Addressing MFE involves refining the math-
Kyle Neal et al. This is an open-access article distributed under the terms of
the Creative Commons Attribution 3.0 United States License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the
original author and source are credited.
ematical representation of the system (Oberkampf, DeLand,
Rutherford, Diegert, & Alvin, 2002).
A related but distinct concept from MFE is model discrep-
ancy. Model discrepancy refers to the difference between the
simulation results obtained from a particular model and the
observed or experimental data. It represents the difference
between the model predictions and the actual behavior of the
system. Model discrepancy can arise due to various factors,
including measurement errors, uncertainties in input data, or
limitations in the experimental setup. Model discrepancy is
typically quantified by comparing the simulation results with
experimental data and can be influenced by both MFE and
other sources of uncertainty (Kennedy & O’Hagan, 2001).
To further illustrate the distinction between MFE and model
discrepancy, let’s consider an example in the context of fluid
dynamics simulations. In fluid dynamics, MFE would refer
to the simplifications and assumptions made in the mathe-
matical equations used to represent fluid flow. For instance,
the Navier-Stokes equations, which govern fluid flow, often
require assumptions such as incompressibility, isotropy, and
neglecting certain small-scale turbulent effects (Reynolds,
1976). These simplifications may introduce differences be-
tween the model and the true behavior of fluid flow in specific
scenarios.
Model discrepancy, on the other hand, would refer to the dif-
ferences between the predictions of the fluid dynamics model
and the observed flow behavior in a specific system. This
discrepancy can arise due to various factors, including mea-
surement errors in data collection, uncertainties in estimat-
ing model parameters, or unaccounted-for physical phenom-
ena that influence fluid flow. Model discrepancy captures the
overall difference between the model predictions and the ac-
tual behavior of the fluid flow, taking into account both MFE
and other sources of uncertainty.
An ongoing research challenge is that MFE cannot be directly
estimated since the true equations governing a real-world sys-
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