Physics informed model error for data assimilation

Jules Guillot; Emmanuel Frénod; Pierre Ailliot

doi:10.3934/dcdss.2022059

Article Contents

2023, Volume 16, Issue 2: 265-276. Doi: 10.3934/dcdss.2022059

This issue Previous Article Comparison of simulation-based algorithms for parameter estimation and state reconstruction in nonlinear state-space models Next Article Harmonic analysis of network systems via kernels and their boundary realizations

Physics informed model error for data assimilation

1.
Univ Bretagne - Sud, CNRS UMR 6205, LMBA, F-56000 Vannes, France
2.
Univ Brest, CNRS UMR 6205, Laboratoire de Mathematiques de Bretagne Atlantique, France

^*Corresponding author: Jules Guillot
^*Corresponding author: Jules Guillot

Received: January 2022

Early access: March 2022

Published: February 2023

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Abstract

Data assimilation consists in combining a dynamical model with noisy observations to estimate the latent true state of a system. The dynamical model is generally misspecified and this generates a model error which is usually treated using a random noise. The aim of this paper is to suggest a new treatment for the model error that further takes into account the physics of the system: the physics informed model error. This model error treatment is a noisy stationary solution of the true dynamical model. It is embedded in the ensemble Kalman filter (EnKF), which is a usual method for data assimilation. The proposed strategy is then applied to study the heat diffusion in a bar when the external heat source is unknown. It is compared to usual methods to quantify the model error. The numerical results show that our method is more accurate, in particular when the observations are available at a low temporal resolution.

Keywords:

Mathematics Subject Classification: Primary: 62M20; Secondary: 62L12.

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Figure 1. Optimization of $ \sigma_{PIME} $, $ \sigma_{QD} $ and $ \sigma_{QSS} $

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Figure 2. Comparison of the evolution of the heat diffusion for the analysis of each algorithm (the more the color is red, the more the temperature is close to zero)

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Figure 3. Comparison of the values of $ X^f_{4} $ and $ X^a_{4} $ for the different algorithms

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Figure 4. Temporal evolution of the estimated temperature of the middle point for each method

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Figure 5. Temporal evolution of the estimated temperature of the middle point for each method with $ dt = 1.5 $

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Figure 6. Global RMSE of each algorithm according to the value of $ dt $

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Figure 7. Confidence interval for the temporal evolution of the RMSE of PIME

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Table 1. EnKF with the physics informed model error.

$\underline {{Initialization}}$: for $ i = 1, \ldots, N $
generate $ w_{1}^{i} $
$ X_{1}^{a, i} = X_{0}+w_{1}^i $
For $ k \geq 2 $
$\underline{Forecast}$: for $ i = 1, \ldots, N $
generate $ w_{k}^{i} $
$ X_{k}^{f,i} = M[X^{a,i}_{k-1}]+w_{k}^i $
$ X_{k}^{f} = \frac{1}{N} \sum_{i = 1}^{N} X_{k}^{f, i} $
$ P_{k}^{f} = \frac{1}{N-1}\sum_{i = 1}^{N}(X_{k}^{f, i}-X_{k}^{f})(X_{k}^{f, i}-X_{k}^{f})^{T} $
$\underline{Analysis}:$: for $ i = 1, \ldots, N $
generate $ \varepsilon_{k}^{i} \sim \mathcal{N}(0,R) $
$ K_{k} = P_{k}^{f} H^{T}(H P_{k}^{f} H^{T}+R)^{-1} $
$ d^{i}_{k} = Y_{k}+\varepsilon^{i}_{k}-HX^{f,i}_{k} $
$ X^{a,i}_{k} = X^{f,i}_{k}+K_{k}d^{i}_{k} $
$ X_{k}^{a} = \frac{1}{N} \sum_{i = 1}^{N} X_{k}^{a, i} $

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Table 2. Parameters values

Parameter	Value
$ n $	100
$ p $	50
$ N $	30
$ R $	$ 0.01I_{n} $
$ K_{final} $	30
$ \alpha $	0.05
$ dt $	1

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Table 3. Optimal values for $ \sigma_{PIME} $, $ \sigma_{QD} $ and $ \sigma_{QSS} $

$ \sigma_{PIME} $	0.016
$ \sigma_{QD} $	0.001
$ \sigma_{QSS} $	0.050

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Table 4. Global RMSE of each algorithm

Algorithm	Global RMSE
PIME	0.017
$ Q $D	0.048
$ Q $SS	0.025

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