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Physics informed model error for data assimilation

  • *Corresponding author: Jules Guillot

    *Corresponding author: Jules Guillot 
Abstract Full Text(HTML) Figure(7) / Table(4) Related Papers Cited by
  • 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.

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


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

    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)

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

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

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

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

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

    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} $
     | Show Table
    DownLoad: CSV

    Table 2.  Parameters values

    Parameter Value
    $ n $ 100
    $ p $ 50
    $ N $ 30
    $ R $ $ 0.01I_{n} $
    $ K_{final} $ 30
    $ \alpha $ 0.05
    $ dt $ 1
     | Show Table
    DownLoad: CSV

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

    $ \sigma_{PIME} $ 0.016
    $ \sigma_{QD} $ 0.001
    $ \sigma_{QSS} $ 0.050
     | Show Table
    DownLoad: CSV

    Table 4.  Global RMSE of each algorithm

    Algorithm Global RMSE
    PIME 0.017
    $ Q $D 0.048
    $ Q $SS 0.025
     | Show Table
    DownLoad: CSV
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