Correction of Model Reduction Errors in Simulations

Transcription

1 Correction of Model Reduction Errors in Simulations MUQ 15, June 2015 Antti Lipponen UEF // University of Eastern Finland

2 Janne Huttunen Ville Kolehmainen University of Eastern Finland University of Eastern Finland Sami Romakkaniemi Finnish Meteorological Institute Harri Kokkola Finnish Meteorological Institute

3 Contents 1. Background 2. Correction of model reduction errors in simulations 3. Examples & results 4. Conclusions

4 Background ACRONYM project at the UEF Aerosols and climate: reduction of uncertainty of the models Aerosol models Formation of cloud droplets Evolution of aerosol size distribution Used in global climate simulations

5 Background Black-box models

6 Background Black-box models Very strict time constraints

7 Background Black-box models Very strict time constraints High-dimensional models

8 Background Black-box models Very strict time constraints High-dimensional models Discontinuities / If-then-else-models

9 Background Black-box models Very strict time constraints High-dimensional models Discontinuities / If-then-else-models No knowledge on true/accurate model

10 Problem description Given Black-box model 1 (fast, approximative) To do

11 Problem description Given Black-box model 1 (fast, approximative) Black-box model 2 (slow, accurate) To do

12 Problem description Given Black-box model 1 (fast, approximative) Black-box model 2 (slow, accurate) A couple of experts To do

13 Problem description Given To do Black-box model 1 Combine models 1 and 2 (fast, approximative) to construct model 3 Black-box model 2 (fast, accurate) (slow, accurate) A couple of experts

14 Cloud droplet formation Inputs: ambient conditions, aerosol size distribution and chemical compositions etc. Output: Cloud droplet number concentration (number of cloud droplets / cm 3 )

15 Cloud droplet formation Inputs: ambient conditions, aerosol size distribution and chemical compositions etc. Output: Cloud droplet number concentration (number of cloud droplets / cm 3 ) Reduced model: Abdul-Razzak-Ghan model Accurate model: Air parcel model

16 Cloud droplet formation Computation times Reduced model: Abdul-Razzak-Ghan model Accurate model: Air parcel model

17 Cloud droplet formation Computation times Reduced model: Abdul-Razzak-Ghan model (0.3 ms) Accurate model: Air parcel model ( s)

18 Cloud droplet formation Computation times Reduced model: Abdul-Razzak-Ghan model (0.3 ms) Accurate model: Air parcel model ( s) Global simulation grid = minutes / timestep

19 Cloud droplet formation Computation times Reduced model: Abdul-Razzak-Ghan model (0.3 ms) Accurate model: Air parcel model ( s) Global simulation grid = minutes / timestep 1 year timesteps about 25 billion evaluations / year of simulation

20 Cloud droplet formation CDNC REDUCED [cm 3 ] CDNC AIR PARCEL [cm 3 ]

21 Approximation errors (AE) in inverse problems Bayesian inverse problems (Kaipio & Somersalo) Additive error term Probability density model for the AEs based on Monte Carlo sampling An approximative marginalization of the unknown and uninteresting AE

22 Approximation errors (AE) in inverse problems UEF // University of Eastern Finland MUQ 15, June 2015

23 Approximation error model Accurate model y = ˆf(x) Approximative model y f(x)

24 Approximation error model Accurate model y = ˆf(x)+f(x) f(x) Approximative model y f(x)

25 Approximation error model Accurate model ] y = f(x) + [ˆf(x) f(x) Approximative model y f(x)

26 Approximation error model Accurate model y = f(x) + ɛ Approximative model Approximation error y f(x) ɛ = ˆf(x) f(x)

27 Approximation error model Accurate model y = f(x) + ɛ Approximative model y f(x) In our approach, we construct a computationally low-cost predictor model for ɛ and use that model in the simulations to predict the realization of ɛ. Approximation error ɛ = ˆf(x) f(x)

28 Example: Cloud droplet formation 1. Construct prior model for x 2. Construct training set (x, ɛ) 3. Train/construct the predictor model for ɛ 4. Construct the final simulation model

29 Example: Cloud droplet formation 1. Construct prior model for x

30 Example: Cloud droplet formation 2. Construct training set (x, ɛ) ɛ = ˆf(x) f(x)

31 Example: Cloud droplet formation 2. Construct training set (x, ɛ) ɛ = ˆf(x) f(x) Here we use inputs x = (x, f(x))

32 Example: Cloud droplet formation 3. Train/construct the predictor model for ɛ Predictor model needs to be

33 Example: Cloud droplet formation 3. Train/construct the predictor model for ɛ Predictor model needs to be fast

34 Example: Cloud droplet formation 3. Train/construct the predictor model for ɛ Predictor model needs to be fast able to handle non-linearities / discontinuities

35 Example: Cloud droplet formation 3. Train/construct the predictor model for ɛ Predictor model needs to be fast able to handle non-linearities / discontinuities Random Forest regression model

36 Example: Cloud droplet formation Random Forest regression model - regression tree

37 Example: Cloud droplet formation Random Forest regression model - ensemble of regression trees...

38 Example: Cloud droplet formation 3. Train/construct the predictor model for ɛ

39 Example: Cloud droplet formation 4. Construct the final simulation model

40 Example: Cloud droplet formation Results: No correction CDNC REDUCED [cm 3 ] CDNC AIR PARCEL [cm 3 ]

41 Example: Cloud droplet formation Results: RF for ˆf(x) RF ONLY [cm 3 ] CDNC AIR PARCEL [cm 3 ]

42 Example: Cloud droplet formation Results: RF for AE 10 4 CDNC REDUCED WITH RF [cm 3 ] CDNC AIR PARCEL [cm 3 ]

43 Example: Cloud droplet formation Results Model Root mean squared Median relative error (cm 3 ) error (%) ARG RF ARG+RF

44 Example: Cloud droplet formation Relative errors

45 Example: Cloud droplet formation Computation times Air parcel model s Abdul-Razzak-Ghan model 0.3 ms RF models about ms

46 Results: Navier-Stokes equations v v µ ρ v + 1 ρ p f = 0 v = 0

47 Results: Navier-Stokes equations v v µ ρ v + 1 ρ p f = 0 v = 0

48 Results: Navier-Stokes equations v v µ ρ v + 1 ρ p f = 0 v = 0 v = ˆf(ρ, µ, v x )

49 Results: Navier-Stokes equations 1.0 Reduced mesh (2.6s) 1.0 Accurate mesh (15.0s) y (m) y (m) x (m) x (m)

50 Results: Navier-Stokes equations

51 Results: Navier-Stokes equations Prior models ρ U(2.5, 5.0) µ U(0.001, 1.0) v x U(1.0, 2.0) 5000 training samples RF predictor model for the errors

52 Results: Navier-Stokes equations

53 Results: Navier-Stokes equations

54 Results: Navier-Stokes equations

55 Results: Lotka-Volterra model Lotka-Volterra model is used to describe the dynamics of prey-predator systems da t dt db t dt = a t (α βb t ) = b t (γ δa t )

56 Results: Lotka-Volterra model Numerical approximation with explicit Euler scheme Models Accurate: t = Reduced: t = 0.25

57 Results: Lotka-Volterra model Prior models α U(0.5, 1.5) β U(0.5, 1.5) γ U(0.5, 1.5) δ U(0.5, 1.5) training samples RF predictor model for the errors

58 Results: Lotka-Volterra model b b a b a a a b

59 Results: Lotka-Volterra model

60 Results: Lotka-Volterra model

61 Results: Lotka-Volterra model

62 Results: Lotka-Volterra model

63 Results: Lotka-Volterra model

64 Results: Lotka-Volterra model

65 Results: Lotka-Volterra model

66 Results: Lotka-Volterra model

67 Results: Lotka-Volterra model

68 Results: Lotka-Volterra model

69 Results: Lotka-Volterra model

70 Results: Lotka-Volterra model

71 Results: Lotka-Volterra model

72 Results: Lotka-Volterra model

73 Results: Lotka-Volterra model

74 Conclusions An approach to reduce the inaccuracies and uncertainties in a simulation model was developed In all tests, the approach significantly improved the accuracy of the models There was only a minor increase in the computational costs Use of the approach with commercial/black-box codes is mostly straightforward

75 Thank you! References Breiman, L., Random forests, Machine Learning 45(1), 5 32, Kaipio, J. and Somersalo, E., Statistical and Computational Inverse Problems, Springer, New York, Lipponen, A., Kolehmainen, V., Romakkaniemi, S. and Kokkola, H., Correction of approximation errors with Random Forests applied to modelling of cloud droplet formation, Geoscientific Model Development, 6, , uef.fi Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M. and Duchesnay, E., Scikit-learn: Machine Learning in Python, Journal of Machine Learning Research, 12, , 2011.